WDW has a fleet of
ECVs that are available for rental at the front of each theme park. Current rate is $50/day, and you can "hop" your daily rental to another park - provided that park has a
scooter available. There is also a deposit required that ranges from $20 - $100 depending on location.
It is true that the theme parks will often run out of rental scooters fairly early in the day. As people go back to their resorts, or to another park in the afternoons, more WDW units are available then. The WDW scooters have just recently been replaced; they are designed to be a heavy-duty workhorse unit (like the ones you see at grocery stores, minus the giant wire cart on the front).
Many people prefer to have an
ECV available to them all the time. Some of the Resorts are HUGE, and the walking there alone can add up really fast! Not to mention if you decide to visit another Resort, or go to Downtown Disney/Disney Springs (which is growing in size daily!)
Walt Disney World has a list of "Featured Providers" (found on
this page)
All of these companies are reputable - and they are the ONLY Orlando companies that can drop off and leave an ECV at your WDW Resort Hotel, so that it is waiting for you when you arrive.
These "Featured Providers" are also the only companies that you are allowed to simply leave the ECV at the WDW Resort hotel at the end of your stay, and walk away, knowing that it will be picked up.
All of the other scooter rental companies MUST meet you to both drop off the scooter, and pick it up when renting on/for WDW property. They are NOT allowed to leave the scooter for you, and you are not allowed to just walk away from it at the end of the rental. Add extra time as appropriate if you are going to use a non-Featured Provider for your rental, for both drop off and pick up.
All of the non-WDW scooter rental entities in Orlando can typically beat Disney's rate by a significant amount, when you look at a multi-day rental. You can rent for any amount of time - from one day to one year.
Hope this helps!