If you go to the disABILITIES FAQs thread near the top of this board, you will find information, phone numbers and links to the places that scooters can be rented.
We recently rented one from Randy's and I can't say enough good about them. They were prompt, professional and courteous. The scooter looked almost new. Randy's is one of the companies listed in the FAQs thread, but all the others listed there are recommended by DIS posters. All of the places will deliver and pick up from your resort, so even though they are not on-site, it doesn't really matter.
For the park ones, they are first come, first serve with no guarantee that you will for sure get one.