I doubt you need a prescription for most vaccines. Typically there's some sort of guidelines that might say normally given to over 55 years old, but there will be some wiggle room due to other hard to prove preconditions such as family history of xxx or sensitivity to yyy. You say you have one of those things and Costco or the public health dept. will give you the vaccine no questions asked. For example when the COVID-19 vaccine was rolled out it was supposed to be for older people, but also people with various other (not easy to test for) preconditions such as asthma. So everyone who wasn't an anti-vaxxer developed asthma.

Medications are different. Need prescription for that, although $$ and a zoom call can solve that problem.