Are you wanting to rent both the shower chair and the wheelchair from the same company?
Are you staying at a hotel? If so, you can call the hotel property in Orlando directly, speak to either the MOD (Manager On Duty) or the Housekeeping department, and ask if they have shower chairs available, and if you can have one waiting in your room when you arrive.
Also, do you use a manual chair outside of WDW? The reason I ask - respectfully - is that sometimes folks thing they can just hop in a manual chair and push themselves all over WDW. They don't realize that unless you use a chair all the time at home (in your everyday life) that you most likely don't have the hand/arm/shoulder strength to go a full day at WDW, let alone multiple days. Even trained athletes at top form can find it exhausting - there are *so* many more hills, swells, ramps, rises, and inclines than you realize... until you are on wheels!
The *average* Guest at WDW travels between 3 & 10 miles
per day; someone unaccustomed to self-propelling a manual chair (especially a rental chair, which will not be nearly as lightweight and easy to maneuver as a custom chair) will find that day two they literally cannot lift their arms to do anything. And that's no way to spend your time at Disney World!
If you have someone traveling with you that can/will push you, then you might let them know to bring along a pair of inexpensive bike or golf gloves (available at any of your finer Walmart stores) to help protect their hands if the rental chair has the old-school hard, black plastic hand grips.
Also, one thing we recommend around here is to take along a length of brightly colored grosgrain ribbon, or a bandana (Walmart also has bandanas, for $1 each, as does Hobby Lobby) and tie it somewhere on your rental wheelchair; it will help it stand out in a sea of similar rentals, and make it quick and easy to spot, especially if a CM has to move it!
I haven't had to rent medical equipment at WDW, but I know that others have, so I will leave the recommendations to them.