We took our daughter to Disney World for the first time when she was two and also did not want the expense of a refrigerator. At that time she was drinking milk but one of her favorite and go-to foods was American sliced cheese, the kind you get in individually wrapped slices. We brought a soft-side insulated lunch box from home and kept it filled with ice we put in ziploc bags. We would buy milk daily in the food court and the ice kept it all cool enough. I wouldn't say it was all COLD but it didn't spoil either. About midway through the trip we ran out of the cheese and it was a panic. We were staying at Dolphin and found packaged sliced cheese at the store at Boardwalk thankfully!
Many times over the years we've filled up the sink with ice and kept soda or beer cold when travelling. My only caution is to remember that the sink probably isn't as clean as it looks or should be so don't eat the ice and don't put food directly on the ice but in some kind of wrapper instead. Also never put ice directly into the ice bucket. You never know what's been in there before and how well it was washed.
On the subject of dining in room, I would also suggest you take a stack of disposeable cups like you'd you'd use at a birthday party or picnic. We never have enough cups in the room for our family and who wants to reuse one once it's dirty?
Good luck.
My friend does a gift card so I might consider that option, but I just love the Disney Dollars, they were so fun to use!
