Yes, it is possible to get bloodshot eyes because of vomiting. Bloodshot eyes after vomiting is eyeball hyperemia caused by the suffocation of back-flow of vein of upper cavity due to throwing up. In other words, when you throw up, plenty of blood flow to the lowest position of the face, the ocular region, as the blood capillary in the eyes is abundant, so with a large number of blood flowing into it, the blood capillary hyperemia occurs. Generally, such case disappears naturally 12 hours later. But I consider that you may vomit due to the stomach and intestine cold caused by catching a cold, taking cold diets or eating much too greasy food, so I advise you to apply hot compress onto your belly and pay attention to defend against cold, avoid eating cold and unsanitary food and pungent food. Wish you feel better soon.