Welcome to Florida, folks. Aligators live all over this state, especially in wetlands and swampy areas and that was what nearly all of Disney property was before Mickey Mouse came along.
While on the Steam Trains tour on my last trip the guide told us some interesting things about gators and Disney. First was back long before the groundbreaking, Walt and one of his close friends and partners in the Florida Project were wearing waders and standing hip deep in swampy water with snakes and gators swimming by. Walt was pointing out where everything would be. They filled in and drained some of the swamps to build the MK. The area towards the back of the park that is wooded where the train goes through alongside a green river (that's duckweed by the way) is a lot like what the MK was until the park was built.
The guide also told us they see gators in that river regularly. Every morning before bringing the trains into the park they stop on the trestle that crosses that river (it's on the spur going back to the roundhouse) and test the low water cutoffs by venting the hot water and steam into the river. Gators often swim up to see what is going on.
There's still a LOT of places on property that gators are attracted to so they stay around. I've heard of them in the Rivers of America and I know that there must be a bunch in the Seven Seas Lagoon and Bay Lake.
Just don't go swimming in the lakes, ponds, rivers, etc. and you should be fine. If you do spot a gator (or other possibly dangerous animals) anywhere they might get dangerously close to people be sure to let a Cast Member know so they can get the guy either back to the water or if it might be a threat have it destroyed.
Destruction is the only way to solve a nuisance gator problem. You can relocate smaller gators safely but the older ones have a wonderful sense of direction and will return to the body of water it lived in before if relocated and become a nusiance yet again.