We just got home at 1am this morning from a one week stay at the Poly and my DD was on Amoxcilin and it had to be stored in the refrigerator and they didn't charge us for the week. So good luck but you should get it for free.
I'm planning on taking a note from my doctor saying that I need to keep my medicine in a refridgerator. I don't know if it will vary resort to resort or from person to person but I figure I'll go prepared. Good luck!
They have some that are about the size of the dorm refrigerators but those are generally the ones that they charge $10/night for. A lot of the ones they let you use free for medications are quite small. Big enough to fit medication in and nothing else.
Like Ducklite said, it was widely abused. So many people were going into the resorts claiming they had to cool down medicine, when they really only wanted a free refrigerator to save the $10/day fee.
When we talked with the people in special services, they told us that the fridges for medication were being switched over from the larger dorm room sizes to little ones about the size of a shoebox. Which one a person gets probably depends on what they have available. It is a wonderful service that Disney provides for all of us who travel with medications that need to be refrigerated(smile).