M.S.
We had them last summer after going camping. My 2 year old dipped everything he ate in the sand and so we came home with them. We had to do the entire family, even though mom and dad didn't have it, to finally get rid of them. My daughter didn't show any signs, and so we didn't do her the first time, and then she had them. Apparently they can be spread from person to person even if there aren't any signs. So, I personally would treat the whole family, and make sure that you are using the proper dose, this includes adults because they have to take more than the kids by a lot. I had Target order the chewable pills for me. It took them a few days, but was SO worth it! I think it was about $20 for the whole family of 4. Then, you need to do it again, even though you don't see any signs of worms. This is the only way to get rid of them.
I know that kids can pick them up in sand, litter boxes, dirt, grass, and any other surface that has been touched by someone who has them. I know that they really like to hang out under the fingernails, so I would cut everyone's nails super short and use a nail brush every time they wash their hands for a while.
I know how frustrating this can be! I feel for you and your little ones. Good luck!