I don't know, I mean, I can see why the babysitter is getting frustrated, but if this is a medical issue and the medicine isn't helping, then perhaps Wendy is right and your daughter is upset to the point of throwing up, so it's more emotionally caused, perhaps the stress of the sitter's punishment is making it worse? Either way, she is not treating your daughter nicely at all and your little girl needs to be in the care of someone who is capable of handling that.
When I think of throw up, there is the sick chunky or liquidy kind, and then there is the little bit when a kid is upset and gags their food out... it sound like she is doing to gagging. Technically, a daycare provider has the right to request children t be picked up for throwing up if they are sick, so it depends exactly what is going on.
Did you sign a contract with the babysitter? If not, then she doesn't have a claim to make you pay, and you shouldn't have to at all. If you have a contract, look for how it can be broken and what the penalties are, do you have to pay if she breaks the contract and to what extent does she have to do that? I'm guessing she can't force you to pay if she chooses to discontinue services.
As for the subsidy, I'm sure it will roll over to a new daycare provider, contact them and see how that works. When you pay the new provider, you will get the subsidy so long as I understand.