There's an off-site clinic that is recommended by the WDW hotels. Just ask at the front desk. I can't remember the name of the place, but it was decent. Expect a wait, however. I had to take DD in for pinkeye and I think we had about a 1.5 hour wait and we had arrived within 45 minutes of opening.
Also, if someone becomes really ill or you don't have access to a car, I've heard here on the boards that the same clinic has a service that will send a doctor to your hotel room for an extra surcharge.