WASHINGTON—The White House gave a politically appointed official the authority to keep aid to Ukraine on hold after career budget staff members questioned the legality of delaying the funds, according to people familiar with the matter, a shift that House Democrats are probing in their impeachment inquiry.

President Trump’s order to freeze nearly $400 million in aid to Ukraine in mid-July is at the center of House Democratic efforts to investigate allegations that Mr. Trump used U.S. foreign policy powers to benefit himself...