The likely culprit is carrageenan, which is added to many commercial plant-based milks to make them thicker and "mucus-like" (like cows milk).
You can try finding one without that.
If you still vomit with a pure milk (like 100% soy or almond), and can otherwise eat soy and almond, then the problem may be psychological (some people have gag reflexes and psychosomatic responses to foods for unusual reasons).
However, I have to wonder why you need a "milk" at all. Is there some reason you need to drink a "milk" rather than water or juice, or some other liquid? What is it nearly impossible to live without it?
(I generally live without it)
Maybe we can help give you suggestions for other beverages, or other breakfast foods (aside from cereal, or whatever demands milk these days).
In either case, you should definitely go vegan