I agree with Karin in that I’ve been training clients for over 18-years, and I’ve NEVER had a client vomit during or after training with me. I work with a wide-variety of clients, including young athletes in outdoor heat environments. I don’t expect vomiting when someone is training with me. I would definitely feel badly, but more importantly I would have a conversation with my client (which I ALWAYS do so perhaps it’s why I’ve not had this problem) about the importance of their being candid with me about 1. how they are feeling physically (are you sick) before training, and 2. checking in with me during the training as to how they are feeling.
I hope that this helps.