J.D. Vance didn’t encounter anyone else with a personal story like his when he arrived at Yale Law School six years ago. His early family life in a poor town in eastern Ohio was tough, with a mother who became a drug addict and a partially absent father. He and his sister spent a lot of time with his grandparents, evangelical Christians who were originally from Kentucky and inspired Mr. Vance to do more with his life.

His experiences are the basis of his new memoir, “Hillbilly Elegy.” The book started out as a quest to answer...