Scupper, thanks for posting those listings of pizza approved locations. The non-chain names are particularly important if they carry the "Rusty Scupper" seal of approval.
One thing I did notice, though, is the Papa Johns location listed is not the one I have used - the listed one comes up in Clermont, Fl.

and one had the phrase "Westgate" which is used to describe Hwy 192/Kissimmee locations. The WDW property is so huge that all the resorts probably aren't served by the same chain location unless the chain has a location in WDW.
All three chain locations that serve the Fort address (4510 North Fort Wilderness Trail, Lake Buena Vista, FL 32830) and I mean Papa Johns, Domino's, and Pizza Hut are closest served by locations out near Hwy 535 in Lake Buena Vista. When you go to dominos.com, papajohns.com, or pizzahut.com and type in the Fort address for delivery, they all give the nearest store as out near Hwy 535 or near Palm Parkway which is a major intersection to Hwy 535.
For Dominos, the closest location using the above Fort address as the delivery address (they will come to the checkin building) is:
8542 Palm Parkway 407-239-1221
For Papa Johns, the closest location using the above Fort address as the delivery address (they will come to the checkin building) is:
11989 South Apopka Vineland Road 407-477-7777
For Pizza Hut, the closest location using the above Fort address as the delivery address (they will come to the checkin building) is:
8699 Palm Parkway 407-239-0205
Chain might run specials as others said. Independants survive because they offer a great product. All I ask is that you tip your driver generously.
Bama Ed
PS - since the OP mentioned this pizza might be for the first/arrival night, we often get the Giddyup And Go take out dinner from Trails End first night (lovingly referred to in our family as the "Gag"). It has 10 pieces of fried chicken, choice of 2 sides from (mac/chees, baked bean, mash spuds, fries - my list may be old), and some big pieces of corn bread. It is PLENTY of food for 4 folks and enough for 5. It costs about $25. It's another idea to put in your hip pocket. 2 pizzas cost me about $25 for my family of 5 so I figure I'm doing well to get out at $5 per head.