I have two crabapple trees that drop crabapples down every year. They get awfully gooey and sticky and make walking through my yard barefoot a terrible ordeal. Is there some sort of enzyme or chemical compound that I can put on them to speed up the rotting process and therefore get them out of my yard quicker?
No, not really.
Fruit trees will drop fruit and if you don't want the fruit on the ground, you'll need to pick them up if you don't have animals that will eat them. Here on my farm the chickens would eat those up, assuming the deer didn't get to them first.
Personally, I'd collect them and compost them - [crab]apples break down pretty well in the pile. Composting is so easy to do that everyone ought to consider doing it.
Here on the farm we use stall forks - like a pitchfork but with closer-spaced tines - to pick up things like this. But raking them into a pike and shoveling them would work too.
I am with J Musser, chickens are the best way. With portable fencing you could corral them all day under the tree, which they will fertilize in return. A pig would work too, but since they are such destructive root hunters, most people bring the food to them.
If you are thinking of picking them up, you might place a tarp under the tree just before the fruit starts to fall. This will be detrimental to whatever is under the tarp, but if its just a bit of grass it might be worth it.
Technically, the answer here is "ethylene".
Ethylene is also an important natural plant hormone, used in agriculture to force the ripening of fruits.
But obviously its not practical to put some lighter-than-air and explosive gas onto your lawn.
Here's one thought though... Can't you just run the lawn mower over them?