The federal voting rights legislation also would have contained funding for election administration processes, including automatic voter registration. Without it, election officials say they will be hamstrung in training staff members and buying needed equipment, running the risk of disruptions. Hundreds of officials from 39 states sent a letter to Mr. Biden on Thursday asking for $5 billion to buy and fortify election infrastructure for the next decade. The letter was organized by a group largely funded by Mark Zuckerberg, Facebook’s founder and chief executive.

Despite that need, at least 12 states have passed laws preventing nongovernmental groups from financing election administration — a wide-reaching legislative response to false right-wing suspicions that $350 million donated for that purpose by another organization with ties to Mr. Zuckerberg was used to increase Democratic turnout. (The money mainly covered administrative expenses, including safety gear for poll workers, and was distributed to both Republican and Democratic jurisdictions.)

Some Democrats and civil rights leaders say they fear that the failure of Democrats in Washington to enact a federal voting law could depress turnout among Black voters — the same voters the party will spend the coming months working to organize.

“Voting rights is seen by Black voters as a proxy battle about Black issues,” said Mr. Paultre, in Florida. “The Democratic Party is going to be blamed.”

In Texas, whose March 1 primary will be the first of the midterms, some results of the sweeping new voting law passed by the Republican-controlled Legislature last year are already clear. In populous counties such as Harris, Bexar, Williamson and Travis, as many as half of absentee ballot applications have been rejected so far because voters did not comply with new requirements, such as providing a driver’s license number or a partial Social Security number.

In Harris County — the state’s largest, which includes Houston — roughly 16 percent of ballot applications have been rejected because of the new rules, a sevenfold increase over 2018, according to Isabel Longoria, a Democrat who is the county’s elections administrator. About one in 10 applications did not satisfy the new identification requirements, she said.

In Travis County, home to Austin, about half of applications received have been rejected because of the new rules, officials said. “We’re now seeing the real-life actual effect of the law, and, ladies and gentlemen, it is voter suppression,” said Dana DeBeauvoir, a Democrat who oversees elections there as county clerk.