I also agree with the previous posters - for the best value rent either a condo or a pool home.
The only place offsite that I would even consider staying without a rental car is at the Downtown Disney hotels, and you'd need two rooms, maybe with the exception of the Doubletree Suites (but only if you have 5 kids, instead of 6). Several years ago I stayed offsite with a friend and used the "complimentary" transportation. It took us over an hour to get from the Magic Kingdom to our hotel 2 miles away.
There are lots of properties in the "Formosa Gardens" area that are literally a 5 -7 minute drive to the Animal Kingdom
(Windsor Hills, Emerald Island, Indian Creek & many others) If you don't mind adding 10 minutes to the drive you can find some very economical pool homes out in the Davenport area. I was helping a friend look for a place to rent, and in that area you can get a nice 4 bedroom home with its own pool for about $700 - $800 for the week. There are advantages of the pool home: it is much easier to keep track of your kids in your private pool - rather than at a resort pool, and you'll have 3 bathrooms; if some kids don't want to swim, no problem you can all be at the saem location, some swim, some watch TV, others play video games - and you can have your eye on all them pretty easily. Here's just one example:
http://www.vrbo.com/129167
If you want specific suggestions for rental communities or a specific home, just post what your needs are, and your budget, you'll get some responses.
I would suggest that you reserve your minivan now if you'll be needing to rent one in Orlando. July is a busy time. You can always cancel the ressie if things change.
Good Luck!
Lisa