Ben, there arent too many options as you can see. There are a few shops (the only one i know for sure is
Kissimmee Pro Dive and given the proximity to Disney, you are likely to be staying within about 10 miles of that place) that will pick you up and take you on a DM guided dive, but i think they charge and arm and a leg for it. Any chance your folks can drop you off or assist you prior to them going to the parks? Unfortunately the two springs mentioned are a bit far north, the other end of Orlando, by 40-50 miles. Any chance your family might consider seeing the wonders of outdoor Florida while they are here - maybe you can all go to Ginnie Springs (about 2.5 hours up the turnpike), they can tube down the river while you go in the cavern and river drift? Another option might be going to Tampa and being dropped off while they go to Busch Gardens? You probably wouldnt be able to hire a car here and i agree with the others regarding taxis - Orlando is just way too big (its at least $2.50 to start and then another 25c per 1/7 mile or something like that, plus maybe a charge for going outside the city limits)
If you let us know what dates you might be here and free, maybe we can help you more if there is a trip already on? Most people are likely to only be free on the weekends though