Hi, Miffy!
Whether or not WDW charges for a fridge for medication seems to just depend on what the CM checking you in decides to do. For a long time, WDW was very generous and people who needed a fridge for medication were not charged for it. Unfortunately, some people started taking advantage of that and so WDW then went to a policy of everyone having to pay for a fridge no matter what the reason. When we last spoke with the people in special services, they told us that the new fridges for medication were about the size of a shoebox and that there was no charge for using them.
I would guess that what type of fridge a person gets for medical reasons will depend on the resort, how many of which kind they have and probably on what phase the moon is in(smile).