We have parents at my center who pull this shit ALL.THE.TIME.
I understand they may not have other options sometimes but the option is you'll just have to take off work. It's bull shit to risk getting a ton of other kids sick, as well as teachers, to pass on to their family members, etc. or in your case, get you and all your children sick!!!
Our policy is if they puke while at the center they have 1 hr to be picked up & cannot return until 24 hours later.
I would def. be typing something up saying they can't come if they are sick because that's super rude & inconsiderate. That's part of being a parent is being there for your child when they are sick AND having to figure something out to take off work, have a back up sitter, etc.