I also have a vegetarian son. We have had to really fight to make sure "vegetarian" doesn't mean "cheese and pasta-tarian." We tell him, vegetarians eat vegetables and not just non-meat junk food. Especially for people like me who are very sensitive to carbs in our diet, a low-protein diet can be a real mistake. She needs to get a good amount of protein and fiber [lentils, beans, etc.] to keep up her energy levels. Also, adequate sleep.
I think what you are doing is great - healthy exercise options, healthy food options. As you already know, there's very little anyone can do to force someone else to eat in a certain way. I've been overweight my whole life and there's really not a thing I can do about it. (I eat healthy, get an OK amount of exercise, and some years I lose and some years I gain. I'm not grossly obese, but it would be nice to be thinner.) It's best for my mental health, I've learned, to not think about weight or calories - weight watchers made me gain weight - but to focus on the positive things I can do (exercise, eating right.) Everyone needs to find their own path.
Focus on the places you can help, as you are already doing. Consider having her talk to a counselor, not about weight, but about her unhappiness with school. Help her to get involved in activities where she may make more friends outside of school (and maybe get some exercise along the way.)
But really - and I think from your thoughtful post you already know this - love her for who she is whatever shape she is. Focus on healthiness and happiness - try as hard as you can in your heart to sincerely ignore the weight/looks issue and be supportive of her at home.