a) I would NOT feel good-clients who vomit are not a badge of honor. A trainer should be able to monitor clients and determine when to increase the intensity or stop the activity (if necessary).
b) I would be concerned about WHY the client vomited. The session may have been too intense but (as Harris and others mentioned), the client may have eaten an hour or less before the session.
A is in the trainer’s court- you should be able to tell if someone looks ill and act accordingly.
B is not directly in the trainer’s ability to control. You can say when to eat and why you leave time between exercise and meals but you will not be with your clients before each session.
Keep your clients (and yourself) safe!