another mom of a "supersize" child here! 3 1/2 year old son, 42 inches, weighs 51 lbs and is solid as a tank, not fat at all.
I bring a stroller for resort/boardwalk/evenings outside the park use, and use a rental in the park. We get a double so 5 yo cousin (and occasioanlly for a free ride, 9 year old cousin) could hop in, but they weren't hopping in enough so we got a single during day, double during evening.
One idea is to go to park w/out a stroller and try to pick one up in the afternoon from someone leaving the park, before they return it. Based on what you pay for a rental, I would be happy to see the stroller get used some more by another family!!!!!
As for a stroller to bring, keep in mind that most strollers are not rated for a child over 40 lbs- you may be fine in one of those but they get hard to push. maclaren strollers are rated to 55 lbs. I just did research on lightweight strollers for disney and bought a Maclaren Volo. You can get a Volo for $120 or less if you get last year's model (PM me I can give you info- I paid $69.99 free ship), it weighs under 9 lbs I believe and has a shoulder strap so you can carry it onto buses with ease. An Esprit Sun Spree is another idea, though I heard the seats are narrower than the volo so I stopped pursuing it- there were great deals on those though!
If your child wants to walk, great- but I have to say, I much preferred the security of knowing my son was in the stroller and couldn't wander off or get taken (it happens!) or knocked over in a crowd!