If you have a problem, you could try the wheelchair rental places at the parks. Another place to try would be the bike rental places that some of the resorts have. Many wheelchairs also use the same size as bicycles.
There is also an
ECV rental place on the Boardwalk,
Buena Vista Scooters, that should be able to help you.
Another possibility that you may want to discuss with your equipment company is wheels that are flat-free. The ones on my DD's wheelchair are foam filled and won't go flat.