Isn't being a mommy hard? We analyze everything, wanting what is best for our children. I'm glad you sought an outside perspective, because hopefully we fellow mommies can help you be at ease with your daughter's voracious appetite!
In my humble opinion, it sounds like the girl simply wants to eat more...and it would be fine to let her. You feed her wholesome foods - it's not as though she is binging on potato chips and soda. A toddler "wild child" burns a lot of energy in a day, plus her little body is still growing at a tremendous rate. She could be going through a growth spurt and feeling a constant rumbly in the tummy! Considering she is two years old and 25% weight, 75% height, she is far from any need to be concerned that she is eating for emotional reasons.
I understand your fear, as you said you struggled with an emotional eating issue. Please be careful to remember that is YOUR issue, not hers. With the predominance of eating disorders hurting our daughters today, steer clear of implanting that idea in her head - especially at such a young, unassuming age.
Coming from a parent of "picky" eaters, be thankful she is a healthy eater!!! Indulge her requests for a while and see how it goes. She will let you know when she is full. I am not a vegetarian, therefore don't know too much about it, but it sounds like her daily diet provides her with the protein she needs (I know when I personally feel insatiable hunger, I need some protein to sustain me).
I hope this helps. Best of luck with your little girl. You seem like a wonderfully conscientious mom and she is lucky to have you looking out for her :)