Also if I may chime in:
One thing we always do is bring along a few of the necessities from home:
Children's Tylenol, Ibuprofen (Motrin), aspirin etc. From what I understand the gift shops on property don't regularly stock children's OTC stuff like those I mentioned. And honestly it has been some time since I have checked (since we end up bringing our own!). Check and see with the front desk if they carry some of these essentials.
There is nothing like having a kid with a fever on property and no car and trying to find these things!
If you need OTC meds there is a Walgreen's that is not very far from property, it is off of 192 (by Sherberth Rd.) So if you don't have a car, and are relying on Disney Transportation, a cab would be needed but it shouldn't be "expensive" (in terms of cab fare) to get these items.
If you are seriously SERIOUSLY ill as the PP said tell the front desk and they can usually send up a Doctor or call Reedy Creek. My Aunt had to do an ambulance ride from the parks to an Orlando ER on their last trip. And said they (Reedy Creek EMTs) were very accomadating, took wonderful care of my cousin who had gotten her finger broke in a bathroom door at the Boardwalk, and in all what could have been a nightmare was actually a rather painless experience.
ETA: I found the address of the Walgreen's we stopped at:7767 W Irlo Bronson Memorial Hwy, Kissimmee, FL 34747. Depending on where you are on property we found it to be very close. There is another one in the exact opposite direction, but since the property is so large again check with the front desk clerks or your cab driver and they (hopefully) should be able to point you to the closest one.
Hope this helps!