Orlando area campgrounds with park shuttles?

Orlando is a big place and the three attractions that you mention are quite a distance apart. You could always rent a car.

We stayed at Fort Wilderness(more expensive) but the convinience of hpping on the ferry was great, as well as being faster than busses and monorail to other sections of the park. We also Stayed at Tropical Palms and drove to each of the locations that you mentioned. The parking was not to bad price wise and the convenience of having your vehicle right there to get back to the campround with half dead sleeping kids was priceless. Kids were the primary reason we choose to park at any destination as opposed to using shutle busses. We always stay till park closings but youger kids almost always don't.

You don't say when you plan to go to DisneyWorld. We lucked up one time and walked in - in January no reservations stayed a couple of nights at Fort Wilderness.
Universal - we got a motel and parked the camper at the motel (it was fairly inexpensive and close to the entrance of the Universal) There are many that fit that category - easier than finding a campsite.

I don't think any have free shuttles to all those destinations but there is mass transit and plenty of taxis. Look in the Kissimmee area

Tropical Palms RV Resort had a shuttle (not free) but convenient. Check recent reviews on www.rvparkreviews.com

I've lived here for 41 years and don't know of any..

Ft. Wilderness of course has the Disney bus and ferry system, but is VERY expensive and almost impossible to book on short notice.