Check out HomeAway.com to check into renting a condo, townhome or house near Disney. Although we didn't take our dog, we rented a townhome about 1 1/2 miles from Disney last summer and it was awesome - much better than staying at a hotel. We paid about the same that we would have paid to stay in a hotel, but we were in a nice gated vacation community in a 3 bedroom/3 bath townhouse with a full kitchen, washer & dryer and our own private screened in splash pool right outside the back door and there was also a nice, big pool in the community. It's especially nice to have separate bedrooms when you have young children so you can put them to bed earlier without disturbing them. We can't do it this year, but I can't wait to go back - maybe next year.
Anyway, go to HomeAway.com and type in "pet friendly Disney World" into the search bar and you can narrow it down from there. Good luck. Hope you find somewhere great to stay and have lots of fun!