And so Mr. Pence became part of a well-established trend among prominent Republicans, who since Mr. Trump rose to lead his party have become regular customers at his establishments. In total, nearly $20 million has been spent at the Trump family hotels since 2015 by various mostly Republican political groups, including Mr. Trump’s own political committees, according to a tally by the Center for Responsive Politics.

Mr. Pence’s stay at the Trump hotel — which bills itself as “a new generation of style and service” — may have been the highest-profile example of a member of Mr. Trump’s inner circle patronizing one of the president’s businesses. But it was far from the first time that a top American official in Mr. Trump’s administration had picked one of the president’s hotels when needing a place to stay or to be seen.

Over all, at least 24 of the 32 individuals who have served in Mr. Trump’s cabinet and 26 of the 53 Republicans in the Senate have been spotted at or spent money at Trump International Hotel in Washington, according to a tally maintained by Zach Everson, who tracks visits to the hotel by foreign dignitaries, members of Congress and other Republicans.

“In a way it is business as normal,” said Mr. Everson, whose newsletter, called 1100 Pennsylvania, examines patronage by the politically connected at the hotel in Washington. “This is the way Republicans are supporting the president, by supporting his businesses.”

Mr. Pence has family roots in Doonbeg, and the president suggested his hotel when he heard that the vice president was traveling there. “It’s like when we went through the trip, it’s like, ‘Well, he’s going to Doonbeg because that’s where the Pence family is from,’” Mr. Short said. “It’s like, ‘Well, you should stay at my place.’”