TT, Is the dog's mouth actually making contact with the children or is it still air snapping? If just air snapping, it could mean a warning... you will have to figure out what it is.
Is the dog lunging at the kids or is it following them then snapping at them? Lunging is an attack mode (which could mean the dog is afraid and is fear biting or resource guarding or has no social manners) following and then snapping is trying to play. You might have to reinforce some play ground rules with both the kids and dog. Things like playing any game where it involves the dog fetching something or chasing something should be avoided. Tug of war games should be avoided as well (unless you can teach the dog that you as the human, will always win), eating around the dog should be avoided as well.
I know you said it happens for no reason now, but is the majority of the snapping occurring anywhere near the dog's food, or sleeping mat/bed? The dog could have developed a guarding resource habit. This can be fixed with some training. The dog also have a hard mouth and if offered a snack or treat will appear to be snapping at the hand giving it. You can train them not to do this but in the meantime, do not let the kids offer Fido any food. ( I have a dog with a hard mouth, and she is simply not allowed to be given food by kids - she has bitten me in her greed to get the food)
Is the dog going for the heels of the children and trying to nip that location? The dog could be just trying to herd the kids. Or they have a strong prey drive and see a running child as prey. This too, can be trained out of them.
Have you tested the dog to see how strong is its bite inhibition is? (a strong one is indicated by the dog releasing as soon as it feels skin) There are about 7 levels of seriousness to a dog bite. The first is air snapping, the second is mouth contact on skin without pressure and immediate release, the third is skin contact with pressure but releasing before skin is broken, the fourth is skin and pressure contact with teeth marks left bit no skin broken but the dog releases, the fifth is skin contact, pressure and skin broken but dog releases, the sixth is skin contact, pressure, skin broken and the dog will not let go, the seventh is skin contact, pressure skin broken, not letting go and dog begins to shake head in order to tear skin.
Levels 2-5 can be dealt with by training, levels 6-7 means the dog is beyond any training and has no bite inhibition. Sadly, most dogs are put down for a level 2-4 bite.
One of the ways a dog communicates is with its mouth. Air snapping is an early warning signal to stay away. It can be from either the dog not liking the immediate circumstances or it's afraid of something. Mouthing is the dog playing (what is important though is how fast the dog releases).
What I do with my dogs is check their bite inhibition. It's something learned from the mother and if the pup was removed too early, it may not have had in reinforced. I aim for my dogs to release as soon as they feel the skin.