It's going to be difficult to train mom and dad because you have allowed the throwing up. Had you nipped it in the bud right from the start they wouldn't have questioned why you were now asking for pickup. It is not sanitary and not something you should have to deal with. If they end up pulling her so be it. Then by the time they go through a few day ares who call for immediate pickup when she throws up they will realize they have an issue and deal with it.

I had a boy throw up once because he didn't want to eat what I was serving. He was sent home. It happened one more time and again sent home. Not sure what was said to him at home but he never did it again.