Privately, several Republican donors have made it clear to administration allies that they want to hear more about policy from a White House where the dominant story line has been Mr. Trump’s unconventional approach to the presidency. Ideally, they would like for Mr. Trump to be that messenger: They note that Mr. Pence may be an explainer, but he is not a decision maker in the White House on policy.

Still, the role Mr. Pence plays has been a comfort to Republicans, particularly those for whom policy is a vital aspect of conservatism.

David McIntosh, the president of the anti-tax group Club for Growth, said Mr. Pence had been crucial in persuading wavering House members to back Mr. Trump’s push to repeal the Affordable Care Act, even though the effort failed in the Senate.

“It goes back to during the campaign,” Mr. McIntosh said of Mr. Pence’s ability to bring calm to policy-focused conservatives trying to make sense of an atypical Republican nominee.

He said that he and other conservatives decided they could support the Republican ticket once Mr. Pence was chosen as Mr. Trump’s running mate. Repeatedly during the campaign, Mr. McIntosh said, his office would approach Mr. Pence to find out whether a declaration Mr. Trump made on the stump would be turned into policy if he were elected.

Mr. Pence, who in 2016 was a bridge between the campaign and evangelical voters, has also been the administration’s liaison at more politically sensitive moments. After the White House drew criticism for its response to the arson attacks at three historic African-American churches in Louisiana, it was Mr. Pence who appeared in front of Mount Pleasant Baptist Church in Opelousas on Friday.