Hi Jamie!
My first thought for you was to say any of the Disney Vacation Club Resorts would be great to stay at because each are so unique and spectacular in their own way. Once I thought more about your children's ages and the fact that you would have at the very least a double stroller and tons of gear, I realized that you need a resort that is conveniently located.
My first choice for you would be Bay Lake Tower at Disney's Contemporary Resort. You are able to use the Monorail to go to Magic Kingdom and Epcot, and you don't need to collapse your stroller which is such a blessing especially after a long day at the theme parks. Chef Mickey's has a character breakfast which your children will love. You also have the option of hiring an in-room babysitter through Kids Night Out, an independent childcare provider.
Both Boardwalk and Beach Club Villas are convenient as well. They both are within walking distance to Epcot and Hollywood Studios or you can take the ferry boat to these theme parks again without having to collapse your stroller. Both the resorts use Kids Night Out as well. Beach Club also has the Sandcastle Club which has supervised activities for children ages 3 to 12.
I have more to say on this subject, but have run out of room. Please send me a follow up question, Jamie and Welcome Home!