The answer to all your questions is yes.
They do rent by the day at all the theme parks, but they can't be reserved ahead of time and are limited in number. If you get there early, you should have no trouble getting one, but by 10 or 10:30, it's very possible that all will be rented out.
You can find out more, including prices and information about offsite companies along with in park, in post 2 of the disABILITIES FAQs thread.
You can find that thread near the top of this board or follow the link in my signature.