I’ve never worked so hard that I vomited (despite the fact it was praise worthy in high school track and field) nor have any of my clients.
If you are well-acquainted with who you’re training, you won’t push them to that point since there is no benefit. So, to answer your question: I would feel horrible if I got somebody to that point; that said, the client has to bear some responsibility in letting you know how he/she is feeling…
The clients that come to me want to enhance their life, and how they travel through it, through movement that is energizing, challenging, dynamic, and thoughtful. Vomiting has no place in that scenario.
I have never seen that in all my 16 years of training clients.
With my personal training clients, I cannot even imagine how it could happen short of somebody already coming in sick and not wanting to let on. Even there I like to think that I would notice something wrong before it would manifest itself in such a drastic way.
As far as group exercise is concerned, I am teaching both water aerobics and MELT. I have seen people in water aerobics sometimes pushing themselves harder than they needed to which may have let them to get dizzy and light-headed but never more than that.
If I had to assume that I was the cause of such misery, I would have to question my ability as a personal trainer.