There are a few in the Lake Tohopekaliga area near Kissimmee (about 30 minutes drive from the Disney area), but these are more about the Airboat and less about the Everglades.
For a true Everglades swamp experience you need to head much further south to the area west of Miami.
That being said, we have enjoyed several excursions on the lake in the Kissimmee area, Boggy Creek airboats is a lot of fun and quite often has Groupon discounts. Wild Willy's Airboats (also lots of fun), Spirit of the Swamp and Orlando airboats are a few others in the area.
These are all on a lake with a marsh area, not in the actual Everglades but the experience is somewhat similar. Wildlife sighting is more hit and miss in this area, our last couple of visits we saw a few young alligators (maybe 18-24 inches long), momma and papa gator didn't seem to be anywhere around. There's also a lot of the native birds and some turtles etc. Book early in the morning or an evening ride if you want to see some of the bigger gators, they sleep during the warm part of the day.
Oh and one tip: gator hunting season starts at the beginning of October, they get pretty scarce at this time of the year (hiding out I guess)