There is some inconsistency in whether or not they charge for refrigerators used for medication. So, do ask, and ask to speak with a supervisor if told no, but don't be too surprised if you are told you will have to pay.
Quite a few people have posted in the past 2 years that they were charged even with a letter from their doctor and showing the medication. Even people who have gone thru the WDW Special Needs Department have been charged. People have posted that when they showed their medication to the CMs, they were told the only way to avoid being charged was to store the medication in the refrigerator at the front desk...even if it meant coming up to the front desk multiple times per day to get doses.
The reason for all this is that some people have seen the "hint" (not on the DIS Boards) that you can get a free refrigerator by saying you had medication, even if you didn't. Because so many people were asking for "medication" refrigerators, WDW started charging everyone, even people who had a legitimate need.
So, good luck.