Vomiting could definitely cause bloodshot in eyes. Many actions with great force will cause bloodshot in eyes easily, for example sneezing, coughing, vomiting and so on. Because the blood vessels are very subtle. Great force might increase the eye pressure, which could break the vessels. That is why we always can find our eyes red after weightlifting, sneezing, and vomiting. Yet it is not a serious situation. Eyes could get clear out soon. Don't worry.
Well, generally speaking, vomiting can lead to a lot of problems on your eyes. First, you should know that Vomiting is typically found in high blood pressure, pain, depression, multiple sclerosis. So when you have vomiting, it is possible to have bloodshot eyes. And also your eyes will be red. But you do not need worry about it, for the redness can disappear in some days. Also, it can lead to retinal detachment too, which is very dangerous to your eyes. So just be careful about it.