Your best bet, if you need a fridge for medical reasons is to actually ring them now and tell them, as they only have a set number of fridges to go around.
Are you booked through a
travel agent or Disney direct? As it makes a difference as to whether they will talk directly to you.
If it is through Disney you can ring them, and they should speak to, or put you through to the special reservations team, who handle all the medical requirements. It will make sure that it is definately sorted out, rather than just put down as a request, which can be ignored if they run out of fridges at the time you are there.
I need one in my room, and when we went last time I made sure that they knew well in advance and it was in the room when we checked in. I have one booked for our trip in October already.
They don't need any proof, as the ADA means they cannot ask for details or proof if you say you have medical needs. They may ask in case it makes a difference to the size of fridge needed, as they seem to have very small ones and also the normal under the counter type size, but if you say it is just for holding medication, they won't ask anything else.
Hope this helps.