We just returned from our trip. We stayed at the Grand Floridian and had two scooters -- a rented one from Walker Mobility for Dad, and Mom's Travelscoot which I just purchased a couple of months ago.
We rented a van from National. I'm Emerald so we were able to pick the van we wanted. They had a Chrysler Town and Country "Stow and Go", which worked perfectly with the scooters. The back seats fold down completely flat, leaving a nice large area for transporting the scooters. My sister also rented a van with National the next day and they had no "Stow and Go" ones left.
So, if you rent from National, be sure to call them and tell them to put in your reservation that you must have a "Stow and Go".
I might add that Mom and Dad took the monorail a couple of times with the scooters, but they weren't terribly comfortable getting on and off the monorail with them. We didn't intend to try the buses and never did, except we took Mom's Travelscoot on the Fort Wilderness bus from the parking lot to Pioneer Hall for Hoop De Doo.
I know others use the buses with scooters and do just fine. We just can't imagine doing it ourselves. We much prefer to load them in the van and drive to where we are going.
I might add we also found the valets at Grand Floridian to be supremely unhelpful and unpleasant. They would see us trying to load or unload the scooters and either stare or walk away -- despite the fact that we tipped each time we left or picked up the van. I think valet parking has become very poor since it was outsourced to a third party. This past trip was a sad example of that. So, don't plan on much help if you valet park.