Guide and service dogs for the visually impaired are the only animals allowed into the Disney parks.
However, if you're travelling with dogs or cats you can still be free to enjoy the parks by leaving them at the resort's own Animal Care Center, a welcoming kennels facility located between the main Disney Parks parking lot and the esplanade to Disney Village, at the top of the moving walkways.
Here, your animal will be cared for with food and water for the duration of your stay. Rates are charged per day and differ for each type of animal.
Note that the Animal Care Centre crew are not permitted to walk dogs, so you should return to the centre during your stay to ensure the animal is given a chance for some exercise.
Your animal must be microchipped and have a valid proof of rabies vaccination issued by an official vet, such as a "pet passport".
Be sure to contact the Animal Care Center directly for further enquiries, questions and advice.