According to what people have posted, there are several different types of refrigerators you might get. The smallest is about the size of a shoebox and the largest is the dorm sized. Housekeeping will provide sharps containers on request.
They should not charge for a frig for medications (especially if you get one of the shoebox size ones), but do check your bill. As dj2 mentioned, the charge sometimes does show up.
A long time ago, they did not ever charge for refrigerators if there was a medical need. Then, people without medical needs started saying they did have needs just to get the frig. At that time, WDW started charging everyone even people who had medical needs because they said that too many people were requesting free refrigerators. That was a few years ago and there were posts where people with insulin or other refrigerated meds were told if they wanted to avoid a charge, they needed to keep their medication behind the desk, even if that meant several trips per day to the front desk to get their medication.
Then, there was a time persiod where some people were charged and others were not. From what people have been posting in the past year, almost everyone with medical needs has been able to get a frig without charge. If you are charged, I'd suggest talking to the manager. And, even if you are told it will be free, check your bill when you check out.