I just wrote a whole paragraph and it deleted :\ but ill write it again. I agree with the others saying to leave it 6 months. Just have her have regular riding lessons for 6 more months. Young children often decide they want something, but a few months later they loose interest. If after 6 months she's still as dedicated as now then ask her instructor to help find her a horse. After a minimum of 2 years riding and handling expereince, if she still wants to ride, look into buying her a horse.
Make sure you get plenty of advice from her instructor.
When the time comes and your in the market for a horse. Be very honest about her riding abilty. So don't look at a horse that says experienced rider and think that she will do ok. Look for a dead broke horse over the age of 15, around 18 would be a good age.
Until then, just keep her going with lessons. Your daughter does not need a horse right now. If I was you id keep her having lessons for at least another year.
If I am not riding I'm thinking about horses, and if I am not thinking about horses I'm riding