If you are lucky enough to see any alligators, enjoy them from a distance. Any alligators you see in Disney World have undoubtedly run across some moron who fed them, and an alligator who associates humans with food is potentially dangerous.
Secondly, from about now until the end of September is hatching season for alligators. If you see a big mound of dried, dead vegetation, about 3 feet high and 5-6 feet across, that's an alligator nest. Stay away from it. You probably won't see her, but Momma is right there within a few feet of the nest, and she's there to protect it.
Also be careful if you see any alligators less than 3 feet long. Baby alligators stay with their mothers until they are about 3 feet, so if you see a 2 1/2 footer, you'll probably also see a few 1 1/2 footers, and then babies 9-10 inches long. The babies are black with yellow banded tails, and the banding fades as they age.
Again Momma's right there somewhere and she'll be real grumpy if you get too close to the babies.
At Everglades National Park, we recommend visitors remain at least 15 feet from alligators, but if I were in a place like Disney, I'd give them at least twice that distance because of the association with food problem.
If you see an alligator, teach your kids Everglades Voices (whispers) and Everglades Cheers (hand gestures instead of squeals and screams).
If they keep a distance that is not threatening to the animals, remain fairly still, and don't make a lot of noise, they can observe alligators for an indefinite period of time. Adult alligators are pretty boring, because they don't do much during the daytime but lay in the sun, but the little guys are really cute. If your kids are quiet, they will hear the little alligators "clucking."
Our DD3goingon23 will sit for an hour watching alligator babies if I let her, and then she'll treat me to a breathless replay of the whole thing all the way home! [EDIT - I should clarify here before someone assumes I'm acting like that Crocadile Hunter idiot -- When Girlie is looking at alligators, she is in my arms...not running around loose!]