In San Jose, Calif., city leaders decided just as a vaccine mandate was taking effect to allow unvaccinated officers to remain employed through the end of the year, with incremental discipline and testing requirements. Mayor Sam Liccardo of San Jose said he wanted to keep as many police officers as possible on the job, but worried about the public health risks of having unvaccinated officers on the streets.

“It’s a huge challenge, and I think mayors throughout the country are balancing the safety imperatives of responding to 911 calls against the safety imperatives of having a vaccinated work force,” Mr. Liccardo said.

But the officers have leverage. Many police departments have an abundance of job openings and a dearth of qualified applicants. And city leaders say they do not want to risk a mass departure of officers at a time when homicides have surged nationally.

“If you decide to move forward with mandating this vaccine, the loss of officers is on you,” Josh Carter, an officer in Leesburg, Va., said at a recent meeting where Town Council members considered a vaccine mandate for municipal workers (and decided not to vote on it that day). “I’m going to come back and ask what your plan is to keep my family and my neighbors safe with little to no officers patrolling our streets or our schools,” Mr. Carter told the Council.

Still, proponents of the mandate have noted that there were risks to the public in not requiring the shots. Mayor Kelly Burk of Leesburg, in the Washington exurbs, said she favored requiring vaccines.

“We have a job, and that job is also to protect, and they’re a component of that protection,” Ms. Burk, a Democrat, said of the town’s police force. “And so if they’re not vaccinated, if they’re not willing to wear masks, then it becomes a real problem.”

Workplace vaccine mandates became more common as the Delta variant sent cases soaring and as President Biden announced plans to require vaccination or frequent testing at large employers. Mandates have succeeded in driving up vaccination rates at health care companies, airlines and other businesses, and relatively small numbers of workers have left their jobs over the issue.