As far as I know, Disney does not allow wagons. You would have one heck of a time getting it on a bus. I have 5 kids, and as much as a double stroller is nice for me, it is a pain to get around with all the people in the parks, fold and carry on a bus, and mine would not fit on the monorail. We always go with 2 smaller strollers. Not umbrellas, but graco makes a really nice literider that folds up with one hand, has a basket and hood. Remember, although baskets are nice and handy all that stuff you pack into them needs to be carried on the bus. Strollers are always a pain to get on the bus, and since I now have twins, that means I fold and carry one and hold a baby and my dh does the same(before at least one of the kids could walk and one was a baby). But I always felt bad for the guy trying to handle one of the big strollers and a family while standing on a packed and moving bus. From experience(many, many Dis trips with kids)some people don't find your family and all their belongings plus a stroller the size of a vw charming. Hope this helps, and relax and have fun.
When we went to WDW last year we found 2 umbrella strollers a must. When we go in December I'm wondering would my DD who'll be five use a stroller enough to bring it? If she'll use the stroller enough that we would want to rent one, I'd bring it. What does everyone else find?