Yet Ms. Harris has promised to bring back the symbolic gesture, as she campaigns alongside former Representative Liz Cheney of Wyoming and makes a play for moderate Republican voters.

The vice president has declined to engage in speculation about which Republican she would choose. “I’ve got to win,” she told Howard Stern when he suggested that Ms. Cheney might be the pick. “But the thing about Liz Cheney, let me just say, she’s remarkable.”

With Ms. Harris naturally holding her cards close, we asked a range of Republican options about their interest level. It is generally considered bad manners to publicly audition for a cabinet post, especially before an election is decided, so many of them demurred or proposed others as better choices.

There are several types of Republicans whom Ms. Harris could pick: those who became liberal stars for opposing Mr. Trump, those who say they will not vote for Mr. Trump, senators who could help shift the balance in the chamber, those who served in the Biden and Obama administrations and those who have been cast out of the party by Mr. Trump.

Here’s a closer look at the possibilities.

Republicans who became liberal stars for opposing Trump

Ms. Cheney, who was functionally exiled by her party after helping lead the House investigation into the Capitol riot, is the biggest Republican name on the potential list. The Harris campaign viewed her endorsement as significant in its effort to win over conservative women in the suburbs who do not like Mr. Trump but are not sold on Ms. Harris.

Ms. Cheney’s spokesman declined to comment and pointed to a September interview she gave at the Texas Tribune Festival in which she said she was “not focused” on a potential Harris administration post. Since then, she has stumped with Ms. Harris and appeared in the campaign’s advertisements.

Ms. Harris has also elevated other like-minded, anti-Trump Republicans. Adam Kinzinger, the former Illinois congressman who served with Ms. Cheney on the House Jan. 6 committee, received a speaking slot at the Democratic National Convention. Former Representative Denver Riggleman, a Virginia Republican, has appeared at campaign events as well.

The letter’s signers include Democrats who are challenging incumbent Republicans in some of the nation’s most competitive races, including Kirsten Engel in Arizona’s Sixth Congressional District; Adam Gray in California’s 13th; Lanon Baccam in Iowa’s Third; and Laura Gillen in New York’s Fourth.

The letter was organized by Defend The Vote PAC, a political action committee that focuses on voting rights and supports Democratic candidates.

The pledge comes as the party is focusing on the issue of democracy in the homestretch of the presidential and congressional campaigns. Vice President Kamala Harris has been campaigning with former Representative Liz Cheney, Republican of Wyoming, who was a leader of the House Jan. 6 committee that investigated former President Donald J. Trump’s attempt to overturn the 2020 election.

“I’m running against somebody who was at the Stop the Steal rally,” said Rebecca Cooke, a Democrat who is challenging Representative Derrick Van Orden in Wisconsin, and signed the letter. “I think right now, voters are concerned about democracy. We need to elect people to Congress who are going to really work to uphold our pillars of democracy and not dismantle them.”

Mr. Van Orden has acknowledged attending the rally that preceded the Jan. 6 mob attack on the Capitol, but he has said he left once it was clear the crowd had turned violent.

Democratic leaders are also pledging to make voting rights legislation their top priority should they retake the House. Representative Katherine Clark of Massachusetts, the No. 2 Democrat, told The Hill this week that a voting rights bill would earn the coveted spot of “H.R. 1” — the designation given to the first and highest-profile piece of legislation at the start of a new Congress — in a Democratic-controlled House.

In 2021, Democrats also made voting rights legislation their top priority, pushing it through the House. But the legislation was repeatedly blocked in the Senate, a fate that would almost certainly await it again.

The legislation at issue would have established nationwide standards for ballot access that aimed to nullify restrictions Republicans have imposed in states around the country following the 2020 elections. Among them are a minimum of 15 consecutive days of early voting and a requirement that all voters be able to request to vote by mail. The measure would also establish new automatic voter registration programs and make Election Day a national holiday.

Democrats would need to capture the presidency and both chambers of Congress to have any chance to enact the legislation. Even then, they would need to overcome the Senate’s filibuster rules, which effectively require 60 votes for any major legislation to move forward.

Senator Chuck Schumer of New York, the majority leader, has attempted to create a carve out to the filibuster rule to advance voting rights legislation, but has so far been unable to muster enough support even among Democrats to do so. He has said he would have a better chance in a new Senate, because two key senators in his caucus who opposed the filibuster change are leaving the chamber.

And the following Sunday, Representative Mike Lawler, yet another Republican on the front lines, took advantage of a break in the Jewish high holidays to tour a kosher shopping center here in Monsey, trying on yarmulkes and posing for selfies.

Where candidates spend precious campaign hours says a lot about who they believe may decide a race. And in the final weeks before Election Day, three of the most endangered congressmen keep showing up at the same places: the rapidly expanding ultra-Orthodox Jewish enclaves of New York’s Hudson Valley.

The investment reflects just how important Jewish voters across the religious spectrum have become in races around New York this year, as war rages in the Middle East and rising antisemitism scrambles political alliances at home.

But in a hyper-polarized nation, ultra-Orthodox voters in particular have emerged as the rarest of swing voters. Not particularly partisan, they have fervently supported both former President Donald J. Trump and Democratic politicians, often acting as a bloc.

How they vote in November could tip several of the nation’s marquee House races — possibly in opposite directions.

“Here you have three congressional seats within miles of each other that could essentially decide the chamber, and the Orthodox Jewish community could play a pivotal role in each,” said Simcha Eichenstein, a state assemblyman and Democratic power broker involved in the races.

The Jewish community is so critical to the outcome that both major party leaders, Speaker Mike Johnson of Louisiana and Hakeem Jeffries of New York, have paid recent visits to its leaders. And both parties have weighed calling in 11th-hour assists from former presidents.

Because Orthodox Jewish voters are notoriously difficult to poll, their influence is unlikely to show up in pre-election surveys, which show the races neck and neck. In many sects, they tend to wait for the endorsements of grand rebbes just days before an election. (None of the candidates in the three races are Jewish.)

The sizable Orthodox Jewish vote has long been a coveted constituency in New York City politics. But while important leaders like the ones Mr. Ryan met remain anchored in Brooklyn, they hold sway over increasingly influential upstate outposts where cheaper land has led to a population boom and new sets of challenges for governments to untangle.

Their impact could be the most significant in Mr. Lawler’s 17th District, home to roughly 30,000 Orthodox voters.

Mr. Lawler won the balance of support there two years ago when he eked out victory over former Representative Sean Patrick Maloney. But his backing was far from unanimous: For example, after former President Bill Clinton helped secure a key local endorsement, Mr. Maloney won 94 percent of the votes in New Square, an insular Hasidic area that Mr. Trump had carried by 99 percent two years earlier.

This time, after securing millions of dollars in federal infrastructure grants and positioning himself as a strong pro-Israel voice in Congress, Mr. Lawler is expected to sweep the Orthodox vote. Political strategists and community leaders in both parties predicted he could win as much as 90 percent of the community, margins so large that they could offset a surge of support his opponent, Mondaire Jones, is counting on from Democratic strongholds.

A couple of dozen miles north in the 18th District, Mr. Ryan has just as aggressively courted voters in Kiryas Joel, home to offshoots of Brooklyn’s Satmar Jewish communities that tend to vote in similarly large numbers.

Mr. Trump won 98 percent of the vote there in 2020 and could clean up again this year. But in 2022, Mr. Ryan, a Democrat, carried about 60 percent of the vote.

Orthodox strategists expect that margin will only grow this year after Mr. Ryan helped secure a $2 million federal grant for a drinking water facility, prioritized securing the community after Hamas’s Oct. 7 attack and, according to The Times-Union, tried to push for changes to the way the state accredits private religious schools, known as yeshivas, a top local priority.

Republicans are not conceding. Mr. Johnson is said to be in talks to meet with one of the community’s top leaders before Election Day, and Mr. Ryan’s opponent, Alison Esposito, has questioned his commitment to Israeli defense. But a member of the community with knowledge of the matter said that Mr. Ryan had already been assured that both Kiryas Joel’s competing Satmar factions would back him.

“To go against the incumbent is like giving the guy a pink slip,” said Rabbi Moishe Indig, a Satmar leader based in Brooklyn. “In order to fire somebody, you’ve got to be bad.”

The same principle could help Mr. Molinaro, whose 19th District includes Hasidic outposts in Sullivan County that could deliver a couple thousand votes. His Democratic opponent, Josh Riley, is being backed by the Jewish Democratic Council of America but has concluded he is better off courting other swing constituencies.

It remains unclear if Mr. Jones has adopted a similar view to Mr. Riley’s, or whether he will try to cut into Mr. Lawler’s apparent advantage.

A former Democratic congressman, Mr. Jones represented much of the district before, and has taken steps to burnish his reputation with Jewish voters. He broke with a former ally earlier this year in an effort to separate himself from his party’s left flank on Israel. He has attacked Mr. Lawler for not condemning Mr. Trump’s comments that “the Jewish people” would be partly to blame if Vice President Kamala Harris wins the presidency.

Yet Mr. Jones has yet to actively campaign here, and there are signs Orthodox voters are still sour over his decision to prematurely stop taking non-emergency constituent casework after losing his seat last time.

“People were upset that he gave up on the community at the end,” said David Lobl, a Democratic strategist who worked on Mr. Maloney’s campaign. “And while he’s trying to make inroads, the Hasidic community has a long memory.”

Mr. Lawler, on the other hand, has been ubiquitous. He visited New Square in June with Speaker Johnson, who called the cause of the yeshivas “an essential fight.” Residents said he has also shown up for ribbon cuttings, bar mitzvahs and home-cooked meals.

“I’ve seen Lawler more times than I’ve seen my rabbi,” said Yossi Gestetner, a public relations executive and a registered Democrat in the district.

Those connections have already paid dividends. Earlier this year, Mr. Lawler’s district director, Rafi Silberberg, relied on Orthodox voters to help nab a progressive ballot line for a candidate whom Republicans planted in a ploy to siphon away votes from Mr. Jones.

Mr. Lawler’s own popularity was obvious last week when he was greeted by Jewish voters shopping for Sukkot in Monsey.

Shoppers pressed in to thank him for helping secure a passport or visa. At least two approached to say that their son had just registered to vote to cast a ballot for him.

One of them, Shmuly Deutsch, said he was used to politicians who only show up around election time. Mr. Lawler had impressed him so much, he said, that he had changed his registration from Democrat to Republican.

“He was the first politician from the Republican Party who actually reached out to the Jewish community,” Mr. Deutsch said. “This time around, I think he’s switched a lot of minds in the community.”

Two days after he made a crude remark at a rally about a famous golfer’s penis size and used profanity to insult Vice President Kamala Harris, Mr. Trump spoke on Monday of the importance of religion in his life, recalling going to church as a child and framing his survival of an assassination attempt in Butler, Pa., as an act of divine intervention.

But even as he courted Christian voters by arguing he was one of them, Mr. Trump shied away from directly mentioning an issue that had previously energized evangelicals behind him but now poses a political liability: his role in overturning Roe v. Wade.

Mr. Trump did cite his appointments of three conservative Supreme Court justices as among his achievements. But he failed to mention or acknowledge that those justices were pivotal in the decision that eliminated the constitutional right to abortion. In a nearly hourlong speech before a crowd that included evangelical leaders, he did not once mention Roe or abortion or use the phrase “pro-life.”

Mr. Trump views abortion as his biggest vulnerability in the presidential election after it hurt Republicans in the midterms in 2022. He pushed to soften anti-abortion language in the Republican Party’s official platform, and he has issued contradictory statements about abortion as he tries to appeal to moderate voters while appeasing social conservatives.

Instead of emphasizing that issue, Mr. Trump focused on portraying his political opponents as antireligious, reviving false and misleading claims that Christians — and Roman Catholics in particular — would not be safe if Ms. Harris were elected president.

Noting that Ms. Harris had expressed support for ending the Senate filibuster, he accused her of wanting to pack the Supreme Court “to overrule your values.”

Under a Harris administration, he warned, “the radical left is not going to leave Christians alone. It’s going to get worse and worse, and you’re going to suffer greatly.”

Mr. Trump has signaled that he believes a large turnout of religious voters could help him win in battleground states. On Wednesday, he will take part in a town-hall-style forum at a church in Zebulon, Ga., an event that his campaign said would focus on religious freedom.

Mr. Trump has chastised Christians for not voting enough, and his event in Concord, a suburb of Charlotte, was an attempt to increase their turnout. Inside a convention center, the atmosphere was a mix of raucous Trump rally and evangelical religious service. At a typical campaign event, Mr. Trump’s crowds will chant “U.S.A.!” and “Trump!” in a display of their fandom. This time, the first group chant during Mr. Trump’s speech was “Jesus!”

At one point, Mr. Trump expressed his wonder at having watched one of America’s most famous ministers, the Rev. Norman Vincent Peale, a best-selling self-help author and pastor who died in 1993. Mr. Trump said Mr. Peale had inspired and influenced him.

As Mr. Trump was wrapping up the story, someone in the crowd called out, “Let’s go Brandon,” a right-wing phrase that stands in for a vulgar insult of President Biden.

Mr. Trump’s overtures to evangelical voters have at times been clumsy. During his 2016 campaign, he drew ridicule after referring to Second Corinthians as “Two Corinthians” during a period when he was vying for evangelical support. This year, his amplifying of a viral video that praised him as the envoy of a higher power sent to rescue America offended faith leaders in Iowa.

Concord was Mr. Trump’s third stop of the day in North Carolina, a battleground state that he won in both 2016 and 2020 and that his campaign sees as central to his efforts to return to the White House.

He began the day with a news conference in storm-battered western North Carolina, where he criticized the Biden administration’s response to Hurricane Helene and made false claims about the federal response.

Later, he traveled to Greenville, N.C., for a rally where he continued to hammer the federal response to the hurricane, lobbed repeated personal insults at Ms. Harris and stoked fear around illegal immigration. He also revived his calls to use the Alien Enemies Act of 1798 to deport criminal gang members without due process. “Think of that, 1798,” Mr. Trump told the crowd. “That’s when we had real politicians that said we’re not going to play games. We have to go back to 1798.”