Hi, Caroline! I know firsthand the specific challenges one faces when trying to continue breastfeeding while on vacation at WDW. At the time of this posting, you can request a small (dorm-sized) refrigerator for $10 (plus tax) per night for your Value Resort room, or you can upgrade to a Family Suite which is already equipped with a small refrigerator. However, many guests use the refrigerators for different purposes including medical reasons. Therefore, it would be difficult to waive the associated fees for any specific reason. For instance, we traveled with a family who has a diabetic child who needed to make sure they had a steady supply of orange juice in their room for middle-of-the-night blood sugar checks, and my dad has to have a refrigerator in his room to ensure that his medication is stored at the right temperature.
When I needed to store milk, I brought along a soft-sided cooler and packed it in my suitcase. Then, as I made bottles, I would fill the cooler with ice from the machines and store the bottles in the cooler and replenish the ice each time I returned to the room. As a tip, place the cooler in the bathtub before you leave for the day as the soft-sided ones may leak as the ice melts. I also had a mini-cooler with an ice pack that I would bring with me to the parks to store already-made bottles. When we were at a restaurant (table or counter service), I would request a hot cup of water to soak the bottle in before giving it to my baby.
I hope these ideas and tips help you plan your upcoming WDW visit, and I hope you have a wonderful day and a magical vacation!