What would your reaction be?

I would be highly annoyed. I would have to calm down before talking to my Sister (or sister in law as it seems to be here). I would tell her that I don't feel comfortable with my Father-in-law to be driving my child or me; so if she has another appointment or anything else that will make her unable to fulfill her duties of driver then she would need to tell me. She would need to tell me in advance so I would have adequate time to make other arrangements for someone else to drive that day. I would also keep a calender so I could mark any dates she has already set. Then I would talk to my father in law about the importance of having my child be buckled in.