When a child is throwing a temper tantrum...it isn't just a temper tantrum. Their body goes under chemical changes and they are under "stress". Also if your child had spitting up issue in infancy it could mean gastric issues (gag reflux) and so something like crying could lead to coughing then to vomiting? As you know stress can do some really nasty things to your body...just an idea! You will have to address the reason why they are having temper tantrums.
Toddler Temper Tantrums - Dealing with Children's Fits & Rages Toddler Vomiting Symptoms And Solutions. Toddler Stress and What You Can Do About It
I really feel for you, the desperation, feeling at a loss...it is a terrible phase and you are not alone. I was there (mine is 2 1/2)...pending to be there again with my second...I think I am stuck there for another 5 years

...haha! You will pull through it and babe will grow out of it - I was told "there is a light at the end of the tunnel". Just keep up the good work. Try to find ways to help stop the tantrums and find the triggers for it. Take breaks and take care of yourself too!
I would also discuss this with your doctor and Good luck!