I always play the "CDC guidelines and district policy for exclusion" card. Sorry, it's out of my hands. If they do not meed criteria for exclusion and the parent sends them to school, my hands are tied
Of course we all know that we have a little more discretion than that, but the black and white method can work for some teachers.
With something ongoing like this, I will usually give the parents a call (with a translator if necessary...my population is about 85% Spanish speaking) and work out a plan with them for when I will call them and when the student will need to go home. For instance, if vomiting is the issue and it has been ruled that they are clear to return, I might suggest to the parent that for a single episode of vomiting we will let the child rest for a while and return to class, but for two or more episodes within an hour, I will call and we will discuss whether the student needs to go home.
While I understand that no one loves a child puking in their class, I think this is a decision that needs to be made between the parent, their physician, and the nurse. The teachers can gripe and moan about it, but I do not think than an inconvenience to them is reason to put a child out of class.