Unless the dog is a seeing-eye dog or some other sort of service dog it won't be allowed at the resort. Disney World does have on-site kennels.
Here is the information about pets, directly from the official Walt Disney World website:
Q. Are pets allowed to stay in my resort room?
A. With the exception of service dogs for Guests with disabilities, pets are not permitted in our Theme Parks or Resorts, or on our Resort and Theme Park transportation. They may, however, stay in one of five on-site air-conditioned kennels, each of which is a member of the American Boarding Kennel Association. And pets are welcome at selected locations within Disney's Fort Wilderness Resort & Campground.
Q. Can you tell me more about the Walt Disney World® kennels?
A. The Walt Disney World® kennels are located at all four of our Theme Parks. You'll find them adjacent to the Magic Kingdom® Park Transportation and Ticket Center, at the entrance to Epcot®, at the entrances to the Disney-MGM Studios and Disney's Animal Kingdom® Park, and near the entrance to Disney's Fort Wilderness Resort & Campground.
Kennel fees are $6 per day and/or $9 per night per animal for Walt Disney World Resort guests ($6 per day or $11 per night for non-Walt Disney World Resort guests). Guests will be required to visit their pets at least once daily. Dog owners will need to walk their dogs at least twice daily (three times daily for puppies). Resort reservations do not guarantee a kennel space for your pets and you should call for availability. Though our kennels were designed for pets such as dogs and cats, we also accept birds, ferrets, small rodents, and non-venomous snakes, if in their own carriers. The kennels are not set up to accommodate exotic animals. All dogs and cats staying at a Walt Disney World kennel are required to have a certificate of vaccination signed by a licensed veterinarian (rabies, parvo, DHLP, Bordetella for dogs; rabies, FPRC for cats).
Q. Can you tell me more about keeping my pet at Disney's Fort Wilderness Resort & Campground?
A. Specific areas of Disney's Fort Wilderness Resort & Campground have been made available to Guests with pets. These are full hook-up campsites. There is a $5.00 per day pet charge. (This is a per day, per site charge only, not per pet.) Sorry, Guests in tents and pop-up campers may not keep pets at their site - they must be kenneled.
The guidelines for Guests with pets are as follows:
Pets will be restricted at all times to the designated pet loops during the Guest's stay.
Pets must be walked on the pet path along the canal only. The owner is also responsible for picking up and disposing after their pet.
Pets may not roam freely in their campsites. The pet must be kept on a leash when being walked or tethered when on the site. This tether can be no longer than 8 feet.
Guests may not leave pets unattended in their site, which causes them to bark or whine, disturbing other Guests.
Pets will not be allowed on pargos (electric carts).
Pet owners not following these policies will be asked to kennel their pet.