My daughter has loose ligaments in her shoulder. Should she have surgery?

Maybe no. The benefits have to outweigh the risk. A good orthopedic evaluation would tell if the shoulder would be restricted more and if the surgery is even necessary. Frequently exercise corrects this.
PT for instability. Most patients who have "loose ligaments" in their shoulder can improve enough with strenghtening exercises (physical therapy) that surgery is not required. If symptoms fail to improve, surgery can be very effective. Be sure to consult with a surgeon who frequently treats shoulder instability. Good luck!
Probably not yet. If she is a teenager or younger, physical therapy for strengthening of the joint is often very effective and may avoid surgery.