Representative Jeb Hensarling of Texas needed a favor: Before retiring, he wanted to anoint a local activist as his successor. Mr. Hensarling, a veteran conservative, reached out to President Trump for help, but the White House hesitated to intervene, according to a person familiar with the overture.

Instead, Mr. Hensarling found a willing ally at Mr. Trump’s right hand: Vice President Mike Pence. Mr. Pence backed the congressman’s favorite, Bunni Pounds, last month in a tweet that blindsided key White House aides.

The eager assistance Mr. Pence provided a senior lawmaker reflected the outsize political portfolio that the vice president and his aides have seized for themselves as the 2018 elections approach. While Mr. Trump remains an overpowering personality in Republican politics, he is mostly uninterested in the mechanics of managing a political party. His team of advisers is riven with personal divisions, and the White House has not yet crafted a strategy for the midterms. So Mr. Trump’s supremely disciplined running mate has stepped into the void.

Republican officials now see Mr. Pence as seeking to exercise expansive control over a political party ostensibly helmed by Mr. Trump, tending to his own allies and interests even when the president’s instincts lean in another direction. Even as he laces his public remarks with praise for the president, Mr. Pence and his influential chief of staff, Nick Ayers, are unsettling a group of Mr. Trump’s fierce loyalists who fear they are forging a separate power base.