Your daughter needs a medical evaluation, this is not normal and likely not because she is a vegetarian. This has been going on for 6 years and you are just now finding out about it???
Sorry, but I find that astounding.
I am not trying to diagnose, but she could have something like POTS. which is a condition that causes drop in blood pressure on standing and walking and can show up as a child, usually around 12 years old, but also younger and can be associated with an infection or chronic fatigue or dehydration among other things. The fact that she is anemic is a red flag and being anemic, means the person has low hemoglobin for various reasons, from not manufacturing enough red blood cells, a bone marrow issue, or a gastrointestinal malabsorption issue that she isn't getting enough iron. Lack of red meat in the diet can be a factor in some cases, but there are nutritional interventions that can easily cover those issues.
Hemoglobin is the "oxygen carrying" part of the blood, so one could be light headed and dizzy with activity and also short of breath.
She needs to be seen by a Pediatrician and have a complete medical work up.