The precise mechanics vary on a state-by-state basis, but in a lot of cases a MD will write a blanket “prescription” that covers a class of patients, and that’s what fulfils the legal requirement for a prescription to dispense/administer a drug. In others, healthcare workers who don’t ordinarily have prescription authority are granted authority to prescribe vaccines by a state law.

The “prescription” here isn’t something the patient will see or be aware of, it’s just part of how the legal requirements around dispensing medications and the logistics of vaccination providers interact.

There’s some information on this from a pharmacist perspective here: https://naspa.us/resource/pharmacist-authority-to-immunize/ .