The vigor with which Mr. Mueller has investigated the flows of foreign money from Ukraine, Turkey and other countries into Washington could be as much a part of his legacy as special counsel as whatever he discovers about possible collusion between Russia and the Trump campaign or presidential obstruction of justice…

All of this has prompted lobbyists to hunt for advice about how to comply with laws governing that sphere, long viewed as toothless. “The phone rings much more often with this question than it did two years ago,” said Tom Spulak, a partner at the King & Spalding law firm who advises on lobbying compliance.

Over the past year, Mr. Mueller and the Justice Department have pursued numerous cases both under the Foreign Agents Registration Act, or FARA, and related to foreign influence operations more broadly. FARA prosecutions were once almost unheard-of, according to a Justice Department inspector general report: For nearly a half-century, from 1966 until 2015, the department pursued only seven.