I've had gift baskets delivered from WDW to a site, I'd bet they would deliver a cake also. Here is the info.
Special Occasion Cakes
You can order special cakes for delivery to your resort room or to be served after your meal at a Walt Disney World restaurant.
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Small decorated cakes (measuring 6"-8", to serve 4-8 people) in chocolate or vanilla can be delivered to your room at any of the Disney Deluxe Resorts. The cost starts at $25 plus tax. The Grand Floridian offers more elaborate cakes at higher prices. Simply call In-Room Dining (room service) from your room phone after arrival.
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Small non-personalized decorated cakes (measuring 6", to serve 4-6 people) can be arranged on arrival at all Disney World resort table-service restaurants. Simply ask when you check in at the podium. The cost is $25 plus tax.
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Personalized decorated cakes can be preordered to be served after your meal at most Disney World resort table-service restaurants. The cost is $25 plus tax for a 6" cake (serves 4), $30 plus tax for an 8" cake (serves about 8) or $45 plus tax for a 10" cake (serves about 16). Call (407) 827-2253 to make arrangements at least 48 hours in advance.