They do not, however there are doctor-on-call services who will make housecalls at WDW. The front desk (at OKW) had leaflets for two of them and the EMTs recommended another service, which we used.
Doctor was prompt, personable, knowledgable. He contacted a pharmacy which was still open at 10 pm and would deliver the drugs he did not carry with him.
Drugs were delivered to the front desk, charged to the room account, and our phone light lit up.
I wish I could remember the name of the service. We had contacted at least one of the other services before calling the EMTs and weren't impressed with the level of customer service they offered.
Can't say enough about the EMTs from Reedy Creek Fire Department either. Everyone was just great!
(I'll edit this with the name of the service we used if I can locate it.)