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?