FORT STOCKTON, Texas—Wildcatter Clayton Williams Jr. made his first fortune in oil. He aims to make another off a treasure buried beneath his family’s West Texas land. It’s a massive trove of water.

The powerful Williams family wants to pipe as much as 25 million gallons a day away from its property at the edge of the Chihuahuan Desert and sell it to oil companies, cities and anyone else with deep pockets and an unquenchable thirst.

The...