Buzz:
I called ABBF, they said they send them to Disney Resorts all the time and even get compliments from the Bell Staff on the way they are delivered. Evidently they have them secured to a small rolling luggage carrier for ease in transporting. No problems reported from any of the resorts.
They did say there was a problem in the parks last summer with jogging strollers, but that was recinded by the Disney rule-makers and are allowing the joggers in the parks once again. What is not allowed are the wagons, and ABBF will tell you that when you go to rent one from them.
HTH!