Are you "POSOTIVE" they were worms and not something he/she ate?? I've never heard of a cat throwing up worms. How old is the cat? and don't worry, it shouldn't be anything that can hurt your or your kids. Even tapeworms (the ones that come out the rear) are the least ones to worry about. In fact, I had a vet tell me the cat can go on for years with tapeworms and never get sick from them in any way but they're very disgusting to us humans of course.
I would call a local emergency vet if your regular vet is close and they "should" be able to give you an idea of what it is over the phone. However, most vets I call only want you to come in and spend the money to get any answers and it wouldn't hurt to look it up online although online sites can almost be more confusing than helpful.
Try not to worry yourself, take him into the vet as soon as you can and see what they recommend. Can you separate the cat from the house for the night or at least until he's seen by a vet?
Good luck, and let us know what he was diagnosed with.