WASHINGTON — Deep into the Senate’s 68-page questionnaire of Judge Neil M. Gorsuch, the Supreme Court nominee was asked to describe how he had come to President Trump’s attention.

The first thing he wrote was, “I was contacted by Leonard Leo.”

Most Americans have probably never heard of Leonard A. Leo, who has long served as executive vice president of the Federalist Society, an organization of conservatives and libertarians who “place a premium on individual liberty, traditional values and the rule of law.” But as Mr. Trump begins the process of filling what could be the most federal court vacancies left to any president in nearly a half-century, Mr. Leo is playing a critical role in reshaping the judiciary.

He sits at the nexus of an immensely influential but largely unseen network of conservative organizations, donors and lawyers who all share a common goal: Fill the federal courts with scores of judges who are committed to the narrow interpretation of the Constitution that they believe the founders intended.

“The Supreme Court needs to be an institution that helps to undergird limited constitutional government,” said Mr. Leo, 51, whose cerebral, unassuming demeanor belies the enormous clout he has developed in Washington.