Mary,
I had a very similar experience with our daughter when she was your daughter's age. During "centers" she would often get to chit chatting (hum, I have no idea where she gets that from

...anyhoo) and during reviews or work time, she'd get bored. That was her exact word "bored". The teacher started giving her more worksheets and color pages, etc. We found that this kept her occupied. Other than the chit chatting, she wasn't having any real behavioral problems.
They had a similar system as what you are describing. They had cards and if they misbehaved, they had to pull a card. After pulling a certain number of cards, you had a time out, notes were sent home, etc.
I am sure that she really does feel sick to her tummy. Poor thing. It's hard to adjust going to school and having a set routine.
I know how you feel about whether or not school problems should flow over to home. Back when this was happening to us, we would sit her down and talk with her about what was going on. We did tell her if she continued to get in trouble at school, there would be consequences.....television time taken away, a favorite toy being taken away, etc. Fortunately for us, the problem went away pretty quickly and we didn't have to do much of the "consequences".
Rest assured, your child is probably not the first child that has done this and the teacher will now how to guide you.