The campaign also intends to organize phone banks, send text messages and run digital ads, including one that encourages New Yorkers to vote for Mr. Biden and Ms. Harris on the Working Families line because “Voting W.F.P. means saying NO to Trump — but also saying New York needs so much more. We need to tax the rich.”

Had the new rules been in effect during the last presidential election in 2016, and in the last gubernatorial election in 2018, the party would have failed to meet the new threshold.

“We are singularly focused on doing the organizing that’s necessary to win and to exceed the threshold and to build the base that comes with it,” said Sochie Nnaemeka, the state director of the New York Working Families Party.

Should it fail, the party risks losing much of its electoral sway at a time when the Working Families Party is facing influential rivals on the left — the Justice Democrats, the political action committee that helped launch the careers of both Ms. Ocasio-Cortez and Mr. Bowman, enabling them to topple congressional incumbents in New York City; and the New York City Democratic Socialists of America, which won several primaries this summer.

Ballot access is the Working Families Party’s defining advantage.

“Right now, given the high degree of activity and enthusiasm that the D.S.A. is generating, if the W.F.P. lost a line, they’d be losing what heretofore had been a significant institutional edge,” said Bruce Gyory, a Democratic political strategist.

For decades, third parties in New York State had to win 50,000 votes in an election for governor to have an automatic ballot line. That began to change not long after the Working Families Party backed the actress Cynthia Nixon as a challenger to Mr. Cuomo in the 2018 Democratic primary.

That choice had significant political ramifications. Just as the Working Families Party was about to name Ms. Nixon as its chosen candidate, unions that are closely allied with the governor withdrew from the party.

When Ms. Nixon lost the primary, Working Families grudginglybacked Mr. Cuomo in the general election, but the damage had been done.

The next year, the state created a commission to look into a public financing system for elections. Part of its legally binding recommendations included a shocker: raising the thresholds for minor parties to qualify for ballot access.

Those recommendations were challenged and defeated in court; the state got around that by tucking them into this year’s budget.

“This was enacted by the public financing commission and passed by both houses of the legislature as part of a comprehensive campaign finance overhaul,” said Richard Azzopardi, a spokesman for the governor. “The conspiracies here are just sad.”

The new rules continue to face legal challenges, including from the Serve America Movement Party, which favors less partisan politics. That party is not backing a presidential candidate in November, so it will summarily lose its ballot access.

Michael Volpe, who ran for lieutenant governor on its line in 2018, said the new rules will most likely stymie the party’s hopes of participating in next year’s New York City mayoral election.

“We could certainly do independent nominating petitions and a write-in campaign, but that effort, as compared to just having the ballot access we thought we won and we thought we had for four years, to me is a deprivation of our constitutional rights and the constitutional rights of SAM voters or potential SAM voters,” said Mr. Volpe, the party’s state chairman.

New York’s Independence Party is backing Brock Pierce, best known for his roles in the “Mighty Ducks” movies, for president; the state’s 28,000-member Green Party is backing the longtime party activist and perennial candidate, Howie Hawkins, for president.

Gloria Mattera, the Green Party’s co-leader, said the new rules amount to “an assassination of the small alternative political parties by the two corporate parties’ cartel.”

Once a party loses its ballot line, it is difficult to reacquire it.

“No one who’s lost it has regained it,” Mr. Gyory said. “When the American Labor Party lost their status, they never regained it. When the Right to Life party lost it, I don’t think it ever regained its status. And the Liberal Party never regained it.”

Not all third parties in New York State are imperiled. The Conservative Party routinely gets more than 200,000 votes on its line. But its chairman, Gerard Kassar, has found common cause with the Working Families Party in its unsuccessful efforts to challenge the state’s new ballot-access requirements.

Mr. Kassar said that their radically distinct politics aside, the Conservative Party and the Working Families Party are part of the same movement.

They share “a belief that the major parties do not always represent the interests of all voters in the state,” Mr. Kassar said.