In Detroit, where county health officials have been running school-based clinics all summer, nurses discovered “strong hesitancy” when they made more than 10,000 calls to parents of students 12 and older to ask whether their children would get the shots and answer questions about them, said the deputy superintendent, Alycia Meriweather. More than half said no.

In Georgia, Savannah-Chatham County Public Schools held their back-to-school clinic at the mall — a “neutral location,” said M. Ann Levett, the superintendent. She is also planning school-based clinics, she said, despite some political pushback and “Facebook chatter” accusing her of “pushing the vaccine on kids.”

Ms. Levett said she was deeply concerned about whether she would be able to keep schools open.

“This is only the second day of school, and already we have positive cases among children,” she said in a recent interview. Her district has a mask mandate, but with 37,000 students, “I just introduced 37,000 more opportunities for the numbers to rise.”