Hello!
The third-person singular simple present indicative form of vomit is "vomits". The present participle of vomit is "vomiting". The past participle of vomit is "vomited".
For example:
"The patient was found vomiting into the toilet bowl".
"The patient vomited into the toilet bowl."
"The patient had been vomiting into the toilet bowl when the sister realized that she wasn't well."
I hope that helps!