I was going to be traveling overseas, and was worried it would be required by a foreign country. I wanted all vaccines handled by my American GP, so the records would be in one place. I was very insistent: "Give me every vaccine for every country."
He said if I wanted the vaccine that bad, I'd have to get it overseas.
It wasn't approved by the FDA for those over 27 until October 2018. Now that it is, I think technically you need a prescription but requesting it through Walgreens/CVS/etc. means they handle that for you. (COVID/flu are similar - if I look at the records for CVS they show a "prescribing doctor" I've never seen.)
Interesting, at the Walgreens portal I don't see any prescribing doctor listed for any of the vaccinations I've gotten there.
In looking at the little pharmacy receipt they give you, there's certainly a prescription number, and a note that says "DR. AUTH REQUIRED" and at the top of the printout there's a field for "DOCTOR" but no value (like there is for "PATIENT"). I guess I have a question for next year, if I can remember!
I did find out that my state has a law that allows administration of flu vaccine by pharmacists without a prescription. That doesn't account for COVID not having one listed, but maybe that's still part of the emergency use authorization?