WASHINGTON—House Democrats have settled on a narrow impeachment inquiry into President Trump centered on his campaign to enlist Ukraine to hurt a political rival, a rapid strategy that will produce hearings within a few weeks and build to a possible vote by November.

That plan was put into action immediately on Friday, when a trio of House committees issued a subpoena to Secretary of State Mike Pompeo for records of interactions between the president and his lawyer, Rudy Giuliani, and the Ukrainian government.

The Intelligence, Oversight and Foreign Affairs committees, which had requested the material about three weeks ago, also scheduled depositions starting next week with five State Department officials, including Kurt Volker, who late Friday resigned as the special envoy for Ukraine, according to a person familiar with the matter, after playing a role in arranging a meeting between Mr. Trump’s personal attorney, Rudy Giuliani, and an aide to Ukraine’s president.