Renting from outside the parks saves time, is cheaper(at least when I rented), guarantees you a
scooter when you want, and allows you to choose your style (three wheels vs. four wheels, size, basket style).
You should be able to find a scooter rental on such short notice still but I wouldn't suggest waiting too long.
The walk from the buses to the scooter rental location is often uphill and much longer if you are in pain. You will then have to repeat your path every day when you leave the park. Then you have to wait in line and fill out paperwork. Uuugghh...
The only perk I see to renting one in the park is not having to wait longer in the bus lines....which is only generally a problem early and late in the day.
I don't know if this ever affects anyone but me, but I have AVN in my ankle and can't stand on a moving bus for any length of time without ending up swollen and in a bit of pain. I always ride the bus with someone in a wheelchair and so I automatically get a seat so it has never been a problem for me. If you take a scooter on the bus though, you will be guaranteed a seat without grumpy passengers getting in your face about asking them to give you their seat and them possibly saying no to your request...you will just have those same grumpy passengers saying nasty things within earshot when you board with your scooter about how you are a lazy, posh visitor who represents everything evil in the world and the downfall of our once great nation.
If you think you might need to ice while in the park, make sure to bring your own bag(s). Restaurants and even food carts are great about giving you ice, but no one had a bag the one time I needed ice quickly to help prevent my corneas from being scratched due to all the swelling.
Hope your injury turns out to be less severe and that you heal quickly!