I have been to WDW a dozen times. I have always required a refridgerator for medicine & never been charged. WDW is quite accomodating. If you get to resort before your room is ready, the front desk will also keep his meds in cold storage for you until your room is ready.
If he needs to carry an injection into the Parks, he can drop off his meds & supplies at First Aid. They will store it in their refridgerator. When it is time for his shot, he can go in & they will give him a private room to do his injection in. They have always asked me if I need any supplies, but I always carry my own. It is nice to know that if I forget alcohol swabs or something, that they can provide them for me.
Disney is very well equipped to deal with medical conditions. It is the easiest vacation destination for me in that regard.