Vomiting when he doesn't get his way

That sounds like a difficult one, vomit is really tough. When he vomits, try to be calm but firm. You could say, "vomiting is gross, and it does not mean you get your own way".  But there is loads of context and nuance missing from the question that makes it a little difficult to proffer advice. I wonder if he has many opportunities to get his own way in cases where that opportunity is safe and possible.

I'm also left wondering by the limited nature of your question, does he get many chances to have a say/be in control? Sometimes a few opportunities to give him control over his situation may help him to feel empowered and some agency and stop this vomiting. Giving children agency supports them in their development as an individual and gives them practice at making decisions, a key requirement of adult life (https://keikiearlylearning.com.au/supporting-childrens-sense-of-agency/). 

I wonder if it's an issue where three year old children suddenly have a voice and want to use it but only sparingly as they are fearful of their new found independence (https://www.verywellfamily.com/3-year-old-developmental-milestones-2764712). I am left wondering if that's the case? If so, maybe you can give him loads of opportunities to give his opinion about issues that affect him (see this paper for some ideas about why (https://link.springer.com/article/10.1007/s13158-019-00252-5). It is sometimes hard to listen to our children's and grandchildren's opinions, mostly because nobody really listened to us as young people. But, to do so, we provide him with loads of experience in offering his opinion, managing his life and give him a sense of being in control in a safe and supported way that may stop the vomiting.

If you are giving him loads of attention, and loads of opportunities to make decisions that impact him ("which shoes would you like to wear today?", "do you want mashed potato or fried potato with your dinner?", "which cup would you like to use?") but aren't earth shattering ("shall we buy you Lego or pay the electric bill?"), you will provide him with a sense of agency that will fulfil his need to be in control and, as you say, "get his own way" without being too much for him to bear.

Good luck!