There is pretty complete information about places to rent from in the disABILITIES FAQs thread (located near the top of the disABILITIES Board, or follow the link in my signature).
The wheelchair and
ECV information is in post #2 of that thread and it includes phone numbers and website links for the off-site companies that DIS posters recommend most often.
Here are the short answers.
The Disney resorts have some manual (push) wheelchairs available at the resorts. They
do put a hold on your credit card while you have the wheelchair. They can't be reserved ahead of time and there is no guarantee they will have one when you arrive or that it will be in good condition or a good size for you. The resorts don't have any power mobility scooters (also called
ECVs) for rent or loan.
The Disney parks have both ECVs and manual wheelchairs for rent at each park. They can't be reserved ahead of time and the parks do sometimes run out of ECVs by mid-morning, especially during busy times.
There is more complete information in the disABILITIES FAQs thread.
You can also rent a manual wheelchair or ECV from off-site companies. There are some companies listed in the disABILITIES FAQs thread that DIS posters have recommended over the years. All of the companies listed there will deliver the wheelchair or ECV to your resort and pick it up when you are done. It is possible to rent an ECV that can be broken down into several pieces to fit in a car trunk. There is information on the different companies websites; let them know you are interested in doing that when making the arrangements for rental.
Power (or electric) wheelchairs are not the same as ECVs.
ECVs are driven using a yoke or tiller that is quite similar to steering a bicycle. There is a separate throttle for forward and another one for reverse.
Power wheelchairs are driven with a joystick that controls speed, forward and backward motion and turning all with the same joystick. Because of this, they are harder to learn to control and most of the rental companies will only rent them to people who have experience driving them (like someone who uses one at home, but doesn't want to travel with it).
Power wheelchairs can't be broken down into pieces and you would need a van with a lift or ramp to transport it if you want to go off-site.