Kissimmee is a great choice and KAZ a very helpful FBO. More info on the airport can be found here
AirNav: KISM - Kissimmee Gateway Airport
Enterprise operate a rental car deal with KAZ but there is also a full branch of Hertz located just next door at Signiture Flight support if you want to organize your own rental. Will be less than 20 mins to Disney anyway.
They did used to have their own landing strip at the park, I have seen it on an old chart. Guess they make more money out of the space for parking.