The day before announcing a new vaccine mandate for employees at religious and private schools in New York City, Mayor Bill de Blasio held a call with eight Jewish leaders to brief them on his plans. The leaders expressed concern about the disruptions that a vaccine requirement could cause at schools and asked for a slower approach, according to two people who were familiar with the call. But the mayor was resolute: The mandate was happening. Mr. de Blasio's announcement of the measure last week -- the largest effort in the nation to force religious and private schools to adhere to a vaccine mandate -- may set the stage for a legal battle, as courts across the country consider challenges to a broad array of mandates to combat the coronavirus.