WDW always used to provide free refrigerators for medication. Then, about 2 years ago, the DIS started getting pretty consistent reports that people were being charged, even if they said it was for medication and even if they went thru WDW RESORTS/SPECIAL RESERVATIONS
(407) 939-7807 [voice]
(407) 939-7670 [TTY]
Many people who reported being charged had previously gotten the fee waived. When they asked during another trip, they were told the policy had changed because too many people were requesting free refrigerators. Several people were told that insulin didn't need refrigeration, so refrigerators would not be provided! The suggested alternative was to keep the medication behind the front desk at the resort, even if it meant several trips per day to the desk.
I think WDW started charging everyone because they were getting lots of requests, including some people who did not have medication, but said they did (I've seen that "hint" posted as a way to avoid paying). Now that word is out that refrigerators are not free, WDW may be getting more lenient again.
So, try asking. If they say no, ask to speak to someone higher up. You could also try the Special Reservations phone number.