WASHINGTON—Attorney General Merrick Garland’s move last month to limit Justice Department probes of journalists is facing an early test from an unlikely source: a federal judge overseeing a criminal case against a talk-show host on Infowars, a right-wing website known for touting discredited conspiracy theories. In an order made public Tuesday, U.S. Magistrate Judge Zia Faruqui said prosecutors refused to answer on the record whether they had complied with the department’s updated media policy when seeking a warrant to arrest the talk-show host, Owen Shroyer, for his alleged involvement in the Jan. 6 riot at the U.S. Capitol. “The events...