Houston has long been the capital of the U.S. energy industry. Now it is vying to be the center of global oil trading, too.

For decades, a small town in central Oklahoma has served as a crossroads for oil markets thanks to its confluence of pipelines and millions of barrels worth of storage space. Cushing, a town of 7,800 people, is the delivery point for one of the world’s most liquid oil futures contract and where prices for U.S. crude are set.

But...