Hi there, Kathi!
Happy birthday, first of all, to your daughter!
Two counter service restaurants in the Magic Kingdom might very well serve your purposes.
Pinocchio Village Haus in Fantasyland features pizzas on the menu. While it's not possible to call out for pizza, you can certainly purchase several of the pizza meals and find some seating inside the restaurant for the celebration.
Casey's Corner offers delicious hot dogs, chili dogs, nachos, and more. There is seating inside and outside; you could certainly grab some dogs and fries and enjoy a little party with a view of Cinderella Castle!
If you want a more traditional restaurant experience - either table service or buffet - you can order a six-inch cake to be delivered right to your table. The cake is $21.00, not including tax and gratuity; no advanced notice is required, and you can order the cake when you arrive at the check-in podium. In fact, The Plaza, The Liberty Tree Tavern, and The Crystal Palace might be able to honor a walk-up order for a cake to take away.
More elaborate, personalized cakes are also available; they can be ordered via the Cake Hotline: 407-827-2253. These require 48 hours advance notice, and delivery locations are limited.
For more information or to make a reservation, give a DIsney Dining a call at (407) WDW-DINE. You can also
book your reservations online.
Have a great trip!
See you later, celebrator,
-Deborah B.