Yes, you can pass out, or faint, from stage fright. When you are stressed or anxious, such as performing in front of people when you have stage fright, then this can make you breath in ways that make you feel lightheaded and dizzy. You may even hyperventilate, where you are breathing rapidly, and take shallow breaths. In this situation, you are not getting enough oxygen due to the shallow breaths you take. And you are blowing off a lot of carbon dioxide with the increased rate of breathing that anxiety can cause. So if you continue to hyperventilate, then you could pass out due to the combination of not enough oxygen getting to your brain, and blowing off too much carbon dioxide.
The other way you can pass out is by a sudden decrease in blood pressure, or a sudden rush of blood from your brain. However, when anxious, most people actually have an increase in blood pressure, as the heart is beating faster and the blood vessels are constricting due to the activation of the sympathetic nervous system and the adrenaline rush from the fight or flight response. But there are some people who respond differently to stress, anxiety, danger, or the sight of injury. In these people, the parasympathetic nervous system is activated, and this slows down the heart and dilates blood vessels. So this in turn leads to a drastic decrease in blood pressure, with the subsequent rush of blood from your brain, and then you faint.
Another way you can pass out from a sudden rush of blood from your brain can occur if you are sitting for long periods of time and then get up too rapidly. The subsequent rush of blood from your head will make you lightheaded and you could pass out. So after sitting for long periods of time, it is advisable to get up slowly.
To counteract feeling lightheaded and dizzy the next time you are to perform in front of an audience, try taking deep breaths and consciously slow down your breathing. This deep breathing will in turn help to provide enough oxygen for your brain, and you will feel the calming effects.