“They just don’t want their kids vaccinated because they bought into the nonsense on the internet,” Mr. Dinowitz said. “And you can’t convince them that they’re wrong.”

Orthodox Jewish leaders have said there is nothing in Jewish law that forbids vaccinations.

Aron B. Wieder, a county legislator in Rockland County, one of the hot spots in the current outbreak, said that while some Orthodox Jews may be “anti-vaxxers,” their opinions are not in line with Judaic teaching.

“There have been many rulings by rabbis and rabbinical boards that have said the opposite,” said Mr. Wieder, who is Orthodox, adding that some have even called getting vaccinated “a religious obligation.”

With more than 700 cases in more than 20 states, the current outbreak is the worst since the disease — which can be fatal in rare cases — was declared eliminated in 2000. It has even crept into the nascent presidential race: President Trump, long a vaccine skeptic, reversed course last week, saying that citizens “have to get their shots.”

In New York, Gov. Andrew M. Cuomo had initially expressed reservations about the bill, saying in an interview earlier last month that the proposal raised a “serious First Amendment issue, and it is going to be a constitutional, legal question.”

But this week, Mr. Cuomo, a Democrat, seemed to shift his position. In a radio interview on Tuesday, he said that he respected the religious exemption, but did not think it applied in the face of the outbreak. “You have a public health crisis,” he said.