As others have said she is lacking protein, possibly iron (and if borderline vegan probably calcium and B12 too).
She really needs to take an interest in her diet and how to be vegetarian, but in the mean time mumsnet can help you with some tips. It is possible to be a healthy vegan, but you need to plug the nutrition gaps.
If she is eating pasta a simple way to make a complete protein is to substitute some of the pasta for butter beans. Or cook some cannellini beans with a little olive oil and garlic until they mush an have that on the side. These beans are often sold canned in tesco/asda 3 cans for a pound. White beans also contain iron.
Tofu is a source of protein and calcium. Some of the flavoured varieties sold in waitrose can just be sliced up and added to the side of a meal. Not so cheap!
I'm guessing you don't make your own pizza, if you do just don't add meat or cheese but give some veggie sausages or the like on the side to make sure she gets some protein. Linda McCartney do vegan sausages.
Drinking a glass a day of soya milk (sweetened tends to be more drinkable) will add calcium, B12 (if fortified) and protein. It is worth considering a B12 supplement (non animal sources are considered unreliable.
Regarding other matter, be careful, you may have lied to her with the best of intentions, but it's an age where lasting damage can be done to the mum daughter relationship. I am not saying that would happen if she finds out, I don't know her, but she may consider it a huge betrayal of trust. As the adult, telling her she broke your trust first (with regards to the fiver) may not wash. Also, be very careful about refusing to buy clothes because she has put on weight, she is not too old for an eating disorder. Again, I don't know your daughter, but 16 year olds I have known, weight can be a very sensitive subject.