Sounds goofy to me. First of all, the carpet cleaning person isn't going to spot clean - he's going to do the whole thing. Secondly, it's a business expense, and she writes if off just like she writes off snacks, toys, a portion of her heat and utilities, supplies like toilet paper and paper towels, and so on. We have in home businesses and we write off a portion of our maintenance and snow plowing, and part of our mortgage.
Your provider should be investing in that stuff (the name of which I can't remember) that schools and camps use to sprinkle on vomit to help dry it up so it can be easily picked up. She should invest in standard spot cleaners and some large bottles of seltzer. Every preschool has a rug area where the kids sit, and sometimes those rugs need to be cleaned and replaced.
I think she's probably guilty of very poor planning. Do you have a contract that you signed that details the prices and any "extras"? Most providers give you a pick-up window of time, but if you're more than X minutes late on 2 occasions, you pay more. They specify fees, days off, paid holidays, an illness policy (like no kids if they are feverish), how the facility will notify you of things like lice and other contagions, and so on. If carpet cleaning is not specifically in there (and I'm guessing you'd have noticed it if it were!), then I think it's part of the cost of doing business. If your kids have been there for 5 years, I can't believe this is the first time someone threw up.
I'm confused that she agreed it was part of the cost of her business, but then said you'd have to pay because it was a large area. What??? Exactly what is the allowable area for throwing up? I think she's just strapped for cash and she doesn't know how to clean up (or doesn't want to) so she's calling in outside help. It's time to have a chat with your provider! I'd also write her a letter, saying that you needed to get your sick kid home and didn't feel that was an appropriate time to discuss it, and also that you have never heard of this "bill the parent" policy and it's not in your contract or verbal arrangement.
I know you don't want to make waves, but since you already pay a premium rate, you don't see why this isn't included. I'd also demand a written contract if you don't have one.