I liked this question on many levels. I am a therapist and have a teenage daughter near the same age and with the same temperament as your own. I also have worked with families in similar situations.
Essentially, at th age of 14-15 most girls are mature and showing signs of developing identity and self confidence. However, at this same age about 25% of these girls are very shy, introverted and quiet, almost to the point of concern. Teachers often say they have great ability but need to participate more, etc.
What is known is that this pattern tends to diminish markedly, and on its own, by 16-17 when maturity takes over. I have seen it many times, both with counseling and without. If the situation is caused by anxiety the counseling helps. In that case it is useful and beneficial to have the assistance of a counselor to help your daughter to overcome fears etc.
However, in many cases this is a maturity issue. It is frustrating for a parent who would naturally worry that their child will be impacted by this introversion and hesitancy. But, as I mentioned, in a few years this naturally takes care of itself, without intervention. Pressure from others seems to make this worse.
What would I do?
I would encourage your daughter to see a therapist one or two times. If the issue is anxiety it will be identified and treated. If not, time will naturally modify this trait and in most situations that is exactly what happens. Your daughter will come out of her shell and be more interactive, but this usually waits a few more years to occur. In truth, I would not be too concerned. I was here myself a few years ago and have been with many families in similar circumstances. If it is anxiety it can be helped, and if maturity...well, that will be overcome on its own. Steven