I would call in room dining of your resort and see. My DS is allergic to eggs and dairy (he can now, thankfully, tolerate small amounts of dairy, very small amounts) and on our
Disney Cruise, they made a birthday cake for my DD that we could all enjoy.
Or, call some restaurants and see.
I'm actually right now checking out the same thing, but this trip is the first one that we're staying off site, so I'm checking out other options as well (my DD's birthday always falls over spring break).
You can always get some Divvies shipped, as they ship nicely packed, but they're cupcakes IIRC. I got those a couple of years ago for a birthday party for my DS, and I was just too lazy to bake the cake that year! Honestly, my homemade ones are better, but it was easy. And, they're not cheap. They do ship well, though.
I'll post back when I find out as well.