The problem you describe is called a subconjunctival haemorrhage and it's completely harmless. It will disappear in a couple of weeks and doesn't require any medical treatment.The haemorrhage is caused by an increase in pressure in the abdominal cavity that spreads to the blood vessels and causes the tiny vessels in the mucous membrane of the eye to burst. This is why the condition typically occurs in connection with coughing or vomiting.
Yes, it is possible for people get swollen eyes after vomiting. This is because of the pressure that is put on the muscles when you vomit. And the pressure also affect the muscles around your eyes, thus cause your eyes get swollen. In fact, some people even get swollen faces after vomiting. Don't worry, having a good rest, The swollen eyes will recover.
Well, it is possible to have swollen eyes. First, you should know that after you have vomiting, it can increase the eye pressure, and then eye strain may occur. In that situation, it will lead to swollen eyes. So you should not drink too much alcohol. By the way, when your eyes have been infected by irritants, your eyes can be itchy. In some cases, it can also lead to swollen eyes. For your situation, I will suggest you drop some eye drops, and it can be effective to the symptom. And having some cucumber under your eyes for a long time, maybe that can relieve your symptom too.