PHOENIX — Nearly two dozen Republicans who have publicly questioned or disputed the results of the 2020 election are running for secretary of state across the country, in some cases after being directly encouraged by allies of former President Donald J. Trump.

Their candidacies are alarming watchdog groups, Democrats and some fellow Republicans, who worry that these Trump supporters, if elected to posts that exist largely to safeguard and administer the democratic process, would weaponize those offices to undermine it — whether by subverting an election outright or by sowing doubts about any local, state or federal elections their party loses.

For decades, secretaries of state worked in relative anonymity, setting regulations and enforcing rules for how elections were administered by local counties and boards. Some held their jobs for many years and viewed themselves not as politicians but as bureaucrats in chief, tending to such arcane responsibilities as keeping the state seal or maintaining custody of state archives.

The aftermath of the 2020 presidential election changed all that.

In the two months between Election Day and Congress’s certification of President Biden’s victory, Mr. Trump and his allies pressured Republican secretaries of state, election board members and other officials in battleground states to overturn his defeat. In a phone call that is now the subject of an Atlanta grand jury investigation into Mr. Trump’s actions in Georgia, the former president urged Brad Raffensperger, the Georgia secretary of state, to “find 11,780 votes” — the margin by which Mr. Trump lost the state to Mr. Biden.

That intense focus on a once-obscure state-level office has dramatically transformed its place in American politics — and the pool of candidates it attracts. Campaigns for secretaries of state this year are attracting more money, more attention and more brazenly partisan candidates than ever before.

All told, some 21 candidates who dispute Mr. Biden’s victory are running for secretary of state in 18 states, according to States United Action, a nonpartisan group tracking races for secretary of state throughout the country.

“It’s like putting arsonists in charge of the Fire Department,” said Joanna Lydgate, the group’s chief executive. “When we think about the anti-democracy playbook, you change the rules and you change the players so you can change the outcome.”

Many of the election deniers are running in solidly red states where it is less likely that their actions could tilt a presidential election. But several others, who have formed a coalition calling itself the America First slate, are running in states won by Mr. Biden in 2020, including in the crucial battleground states of Michigan, Arizona and Nevada.

The coalition’s members are coordinating talking points and sharing staff members and fund-raising efforts — an unusual degree of cooperation for down-ballot candidates from different states. They are in strong position to win Republican primaries in those battleground states, as well as in somewhat-bluer Colorado and heavily Democratic California.

Their chances in November, should they succeed in the primaries, could rest heavily on how well Republicans fare in the midterm elections, given voters’ tendency to vote for down-ballot candidates such as secretary of state from the same party as their choices for governor or senator.

While local election officials typically oversee the counting of individual ballots, and state legislatures sign off on slates for the Electoral College, secretaries of state often certify elections and set the tone of how elections are run. Their election-management duties generally include distributing voter registration cards, allocating voting machines, educating voters, auditing election results and ordering recounts.

Had secretaries of state taken their cues from Mr. Trump in the last election, they could have put their thumbs on the scales of fair elections by forcing the closure of polling places, removing ballot drop boxes or withholding other resources that could make voting easier in heavily Democratic precincts. Worse, critics say, they could have raised doubts about, or even refused to certify, Mr. Biden’s victories.

The powers of secretaries of state to subvert elections vary from state to state and are largely untested in court. Mr. Trump’s phone call to Mr. Raffensperger in Georgia raised the specter of out-and-out fraud in the tabulation of a presidential vote. Short of that, in states where secretaries of state have the power to certify elections, the refusal to do so could be a vital step in overturning one. In a presidential election, state legislators and the governor hold the power to approve an alternative set of presidential electors, and refusing to certify could boost such an effort.

In contests for governor or for House or Senate seats, the refusal to certify the result of an election could send states into uncharted legal waters.

Those who say they are alarmed at the possibilities include many current Democratic secretaries of state — and a few Republican ones.

“The narrative that is being promoted by people who are ill-informed and simply trying to promote a political narrative to benefit themselves in a particular candidacy is very dangerous,” said John Merrill, the Republican secretary of state in Alabama who is term-limited.

The significance of the America First coalition’s parallel efforts can be seen clearly in Arizona, where the slate’s candidate is Mark Finchem, a former firefighter and real estate agent who has served in the state House since 2015 and has become the leading Republican contender for secretary of state. He has raised some $663,000 for his campaign, according to state filings, more than the two leading Democratic candidates combined.

Mr. Finchem, who declined to comment for this article, was in Washington on Jan. 6 and attended the Stop the Steal rally that led to the storming of the Capitol. He has publicly acknowledged his affiliation with the Oath Keepers, the far-right militia group whose leader and other members were charged with seditious conspiracy for their roles in the Capitol riot. He championed the Republican-ordered review of the 2020 vote in Maricopa County — though he never endorsed its conclusion that Mr. Biden won — and received a prime speaking spot in Mr. Trump’s Jan. 15 rally outside Phoenix.

There, Mr. Finchem told the crowd that the 2020 election had prompted him to run for secretary of state, said he was part of a “nationwide populist movement to regain control over our government” andcalled for the State Legislature to decertify the presidential result in Arizona, which Mr. Biden carried by nearly 11,000 votes.

Ladies and gentlemen, we know it and they know it — Donald Trump won,” Mr. Finchem said.

The coalition’s other candidates include Jim Marchant in Nevada, a former state legislator; Rachel Hamm in California, who contends that Mr. Trump actually won that deep-blue state; and Kristina Karamo in Michigan, who developed a high profile in conservative media after she made uncorroborated claims that she had seen fraudulent ballots being counted in Detroit during the 2020 election, allegations that have been disproved by both local election officials and courts.

Major donors to the coalition include such promoters of election conspiracies as Mike Lindell, the chief executive of My Pillow, and Patrick Byrne, a former executive at Overstock.com, both of whom have also helped fund several election-denial campaigns and lawsuits. Mr. Byrne said he gave the group $15,000.

“​​We would like as many like-minded secretary of state candidates to come forward as we can,” Mr. Marchant said at a Las Vegas conference that featured members of the coalition along with speakers who are well-known to followers of QAnon conspiracy theories. “I’ve got a few that have contacted me. We’re working to bring them into the coalition.”

In an interview, Mr. Marchant said the group had presented its theories about the 2020 election at three “summits” in different states recently and planned others in Wisconsin, Texas, Colorado and Nevada.

He brushed off concerns about undermining confidence in elections and instead assailed sitting state and local officials for resisting further audits of the 2020 vote. “If they’re so confident, wouldn’t they gloat and say, ‘See, we told you so?’” he said. “They won’t. They can’t afford to do that.”

Tony Daunt, a longtime Michigan Republican official who was appointed last year to the panel that certifies the state’s election results, said Ms. Karamo, who has falsely claimed that Mr. Trump won Michigan, was unqualified to be secretary of state because of the “nonsense regarding the stolen election.”

But Mr. Daunt and Mr. Merrill, of Alabama, are among very few Republican election officials who have publicly criticized the spreading of lies about the 2020 election. Instead, pro-Trump Republicans are enthusiastic about those candidates, and both the candidates and their supporters say the changes they are pushing for will make it more difficult to commit election fraud, which they portray as a pressing threat.

Mr. Finchem is sponsoring a bill in Arizona that would treat all voters’ ballots as public records and make them searchable online. Another of his bills would require all ballots to be counted by hand, although studies show that hand counting introduces more errors. And he has repeatedly called for “currency grade” paper as a countermeasure against fake ballots, though there is no evidence that fake ballots have posed a threat to fair elections.

Nothing and no one has catalyzed Republican enthusiasm for secretary of state contests more than Mr. Trump himself, who has offered three endorsements for Mr. Finchem, Ms. Karamo and United States Representative Jody Hice, who is challenging Mr. Raffensperger in Georgia’s Republican primary. Mr. Hice reported more than $575,000 in donations for his secretary of state candidacy in June, twice Mr. Raffensperger’s total.

And Mr. Marchant, in Nevada, said he entered the race after being encouraged by allies of Mr. Trump.

While the money being spent on races for secretary of state as yet does not approach the fund-raising by candidates for governor or Senate, they are no longer the low-budget affairs they once were. In Georgia, Michigan and Minnesota, fund-raising is more than double what it was at this point during the 2018 midterms, according to the Brennan Center for Justice.

Despite their fund-raising struggles in the Arizona contest, Democrats are having some success creating a national support structure for secretary of state candidates.

Jocelyn Benson, the Democratic secretary of state in Michigan who is facing a likely re-election battle against Ms. Karamo, has raised $1.2 million this campaign cycle, more than six times what her Republican predecessor raised by this point in 2014. Nationally, Democratic candidates for secretary of state raised six times as much money in 2021 — and from five times as many donors — as they did in 2017, according to ActBlue, the Democratic donation platform.

Jena Griswold, the secretary of state in Colorado and the chairwoman of the Democratic Association of Secretaries of State, said she had hired full-time staff for the first time in the association’s history. She said the group had set a fund-raising goal of $15 million for this cycle, far surpassing the $1.8 million it raised in 2019 and 2020, and had raised $4.5 million toward that goal so far.

“The stature of the office is different, and the stature of what officeholders are doing is also different,” Ms. Griswold said.

Susan C. Beachy contributed research.