In 50 years of making bourbon barrels, no one had ever offered Leroy McGinnis more than what he charged for them. But over the past six months, multiple distillers have offered to pay him $250 a barrel—a 70% premium above the $150 list price.

The offer illustrates just how scarce bourbon barrels have become. As bourbon sales have soared, both barrel production and the lumber industry have struggled to keep up.

Mr....