According to what people have posted recently, it's very inconsistent whether or not you will be charged for a refrigerator for medical reasons. It used to be free of charge for medical use, but (as tjmw2727 posted) there were too many people who were claiming they had a medical need when they didn't just to get the fee waived, so they started charging everyone (even people with legitimate medical needs). Recently, people have been posting more often that they were not charged.
I don't have access to the number right now, but you could check with WDW Special Reservations about a refrigerator (they are the department that deals with special needs). They might be able to get one at no cost (although some people have not had luck even thru them). I'll post the phone number when I get home if no one else has.
Also, you might want to check with the Pharmacist to find out whether your medication needs to be refrigerated or just kept cool. Be careful if you use and ice bucket and keep your medication on the ice. Some people have found frozen medication because contact with the ice was a little too cold.