Although not in Austin my daughter usually once a year competes at the Hog Heaven Shotgun range near Dripping Springs and nice facility. Only 22 miles from Austin to there.
Hog Heaven is a world-class, state-of-the-art private sporting/shooting club in Dripping Springs, TX. Adjacent to the sporting club is a special events venue for parties, weddings, corporate, and other special events that is second to none.