You can order a cake at almost every sit down restaurant. They keep a stash in the back. Ours was chocolate with chocolate icing and a sugar Mickey Happy Birthday on top. It is $12.50 and they will bring it to your table (for that size). I recommend calling a few days in advance to confirm and remind at check-in.
I don't even like cake and it was awesome. We took ours back to the room and ate it all week. We said we were going to get a cake at our first dining every trip just to bring it back to the room and fullfill our late night cravings once our DD has passed out.
The Disney delivery charges are very high. You can also pick up at cake at the Boardwalk Bakery or the Grand Floridian bakery. More choices in ordering there compared to the restaurants.
good luck.
Karen