Wow. What an unusual situation.
I once had a child in my group who sometimes had huge crying bouts due to separation anxiety, and would throw up because of all the mucus she'd ingested.
I have to be really honest here, SLM, and say that there are a couple things that raise some questions for me. I'm not 100% sure if they are true red flags-- I think you will know that in your gut or not--but here's what stands out to me. First, and foremost, they haven't taken her to the doctor. Why not? What reason would a parent have for not taking their chronically vomitting child to the doctor? This suggests to me that there's a level of parenting awareness that they haven't yet attained, or that there's some stresses at home that they aren't sharing with you. I couldn't guess as to what.
And this falls hand in hand with this little girl having had so many caregivers. Either they are really unaware of how this is affecting their daughter, or they are hiding something and leaving caregivers in order to cover up for their own mistakes. It may be that they are hiring lousy candidates and are slowly figuring their way through finding good care. In any case, this and the fact that they are so willing to go with a less-healthy way of feeding youngsters, whose bodies need snacks, suggests that there's something lacking in experience.
I think the doctor's note is a great way to point them in the right direction. Someone needs to insist that this child get medical care for this before she returns to your care. Give them a short amount of time to do this, and tell them by state guidelines, you can no longer accept her in your care without a doctor's note. There's a 24 hour exclusion policy for vomiting in Oregon, and kids can't return to school until 24 hours after the last bout of vomitting. You could be violating state law in accommodating them, because she's *never* past that amount of time if she's vomitting each morning. She should have been checked long ago.
H.
I want to add: if the parents do decide not to take their daughter in, please call your local child welfare division. There may be something terribly amiss, and a home welfare check would be in order at this point. Not seeking medical treatment for a child with these symptoms is just worrisome to me.