ABHA, Saudi Arabia—Arwa Alneami wanted to be an artist ever since she was a child. But growing up in the conservative region of Asir, she was constrained by a rigid strain of Sunni Islam that has long defined life in Saudi Arabia and the kingdom’s image to the outside world. When she drew a bird, Ms. Alneami recalls, teachers would scold her and cross off its head, saying only God can create life.

Now that religious control is coming under its sharpest challenge in modern times. Saudi leaders, spurred by the need to diversify…