We have 4 and 6 year old daughters, who regularly climb to the top of Stone Mountain, so they're in good shape, but we ALWAYS bring our double stroller to Disney, because there are time you want to hustle up a ramp or through crowds and it's much easier when you can ferry the kids on the stroller rather than making them hurry-they get so distracted and it's easy to lose them.
The stroller is also good for packing lots and lots of water in the baskets underneath, and sometimes one of the girls will just crash at some point and konk out for about 15 minutes in the stroller. We usually bring a big towel along to either keep the sun or rain off of them if they've had too much.
We've had a graco double wide stroller for the last 5 years, and we easily pack approaching 100 lbs on it with a 35 lb girl, a 48 lb girl, and easily 10 lbs of water and 10 lbs of assorted junk. Never had a problem with it not taking the weight.
It fits without folding in the handicapped section of the monorail, so if it's 10 at night and both kids are asleep, you can roll it onto the monorail without making them wake up. THe stroller is supposed to be the exact width of a wheelchair, I think it may be an inch or two wider but it fits through all the doors. When it's really loaded down it takes some arm strength to manuver it around stuff, but it rolls like a dream.
Might want to hit some garage sales in your area to see if you can find a used Graco double stroller. Mine's navy blue.
Also, if you find one that's kinda grimy, you can unscrew all the little screws and white nuts and throw every piece of fabric on the frame in the wash-we've done this before every trip to make sure we've got a clean stroller. We also grease the wheels but I admit we're going a bit overboard there...