Can microneedling with PRP cause facial hair growth in a woman?

This is a good question.  Microneedling with PRP should not cause facial hair growth.  Facial hair growth in women is caused by several things;  genetics, hormones and age being the main culprits.  You may find that as you age, you will see more hair on your face.  There are many options for permanent hair reduction.  By treating facial hair now, you may prevent it from becoming an issue later on. 

Platelet rich plasma is a diverse therapy that is used both for the face and scalp. This tends to get some patients nervous about using the treatment on their face as it promotes hair growth on the scalp.
What causes facial hair growth in women?
There are three major culprits; one being hormonal imbalances, the other being age, and the last being genetics.
Every female has a few androgens present in her/their body. There won't be any drastic changes in facial hair growth as long as the estrogen levels are higher than the androgen levels. If and when these levels undergo an imbalance is when vellus hair or 'peach fuzz' becomes coarse in some areas.
When a female goes through womanhood, there are multiple changes that occur throughout their life that can also lead to facial hair.
Can platelet rich plasma cause facial hair growth in women?
The purpose of using platelet rich plasma on the face is to promote the production of elastin and collagen. Therefore, the short answer would be no.
What is a permanent way to stop hair growth?
A very good option to make hair follicles redundant would be laser hair removal.