Yes, it is very normal. You don't have to worry about it at all. Tear is very important to our eyes. It is composed of water, protein, mucus, anti-bacterial agents and so on. It could lubricate our eyes so that we can rotate our eyeballs without discomforts and grittiness. The anti-bacterial agents could kill 90% of bacteria in eyes to control infections. Meanwhile it can also nourish the eyes with protein. When we are coughing, sneezing and vomiting, we all tend to have watery eyes and bloodshot in eyes.
Almost all of us will get watery eyes when we vomit. Because our organs are connected, and our head is downward, those liquid possibly stimulate our glandula lacrimalis. Then we will tear.
Vomit can cause rhinorrhea, and cough as well. As I said they are all connected.
So, what you need to do is avoid vomiting next time. By the way, clean your mouth thoroughly after vomit.