Yes, it is true that the vomiting can cause black eyes. When you vomit, your intestinal part will be damaged. Your immune system in the whole body will be affected. You may find that the eye nerves will be affected. The black eye can be suddenly caused by the sudden increase blood circulation and strong eye pressure. You'd better go to see the doctor and find the treatment if you have the black eye because of vomiting.
Vomiting is one of the most harsh activities we might do, and it would lead to bleeding under your skin, and due to a lack of wound, the blood would stay around your eyes, that is black eyes. But this is really rare. Also, black eyes could be seen as a sign of weak body, which needs to be taken care of .