Hi and
to both DisBoards and disABILTIES!
If you are in anything but a Value Resort there will automatically be a refrigerator in the room. If you will be staying in a Value Resort just explain, on the reservation, that you need a refrigertor for medications.
I have never had any problems in any hotel (Disney or otherwise) with medications being left in the room. If you feel strongly about it, for those which do not require refrigeration you can put the medicines in the room safe.
In most hotels, and all of Disney's, there is an electronic record showing the time and key which was used to enter each room. A housekeeper's key will only work in the rooms that are assigned to a particular housekeeper. An a manager's key, for example, can be quickly cancelled if it is reported lost or stolen.
So it is very unlikely that someone will steal your medications. Also, I do not pring the containers with me (except for liquids). I have weekly pill boxes, and put each day's pills in the correct place. I have separate boxes for the pills in the morning and the evening. So even if someone looked at them they would not know what they are.