Shoulder injuries can be hard to diagnose. A good starting point is an exam by an orthopedic surgeon and an X-ray. In fact, a series of X-rays may be needed to get the right angle to see everything.
Major trauma of this type has been linked with a shoulder injury called a floating shoulder. With a floating shoulder there is a fracture of the collarbone and the shoulder blade. The shoulder socket can actually detach and move forward and down. That’s what can give the arm a drooping look.
Or sometimes when the arm has been pulled like you described, a traction injury occurs. This means the nerves got stretched. Nerve damage can result in a drooping arm, too.
Early treatment is always advised for the best long-term result. Working with horses, your sister will need arm strength and motion. This may be the best way to convince her to see a doctor sooner than later.