I've been to many parks in Orlando and have found their all a little different when it comes to handicap policies, wherever you get to, find guest relations and they will help you.

Not Disney, sorry...I recently went to Universal- decided at the last minute I wanted a scooter so I called Guest Relations the day prior and luckily they had one. I was told they can go quick at times. Had no trouble the day of, picked it up just inside the gate. If I remember from my research, some are first come first serve, and some are reservation- style.