There's a medical term for vomiting which is "emesis" but "throw up" would probably be the right choice. As opposed to emesis, which might confuse the person hearing or reading the word, throwing up is relatively polite and not too clinical.
For as many ways as there is to describe getting drunk, plastered, snockered, blitzed etc. There seems to be twice as many ways to describe "losing your lunch." -- the act of throwing up.
For your amusement, here's a short compilation of these terms shared by Thecareandfeedingofman.com.
And here's A Definitive Ranking Of Every Slang Term For Vomiting from BuzzFeed.
Tip! Food poisoning, infections, and motion sickness can all cause vomiting. Read more about the causes and learn the right treatment to stop the vomiting at Parents.com.