“We’re going to be running for re-election or election, and what’s the vision?” said Representative Scott Perry, Republican of Pennsylvania and a Freedom Caucus member. “When you make that investment of your vote, you want to know who is going to be embodying that vision. Who is that person right now? It’s not Paul Ryan.”

Whoever takes the helm will help determine the ideological direction of a conference that remains every bit as fractured as it was when Republicans drafted Mr. Ryan — a former vice-presidential nominee — to succeed John A. Boehner, who quit in the fall of 2015 rather than face a conservative rebellion.

Representative Matt Gaetz, Republican of Florida, who describes himself as a Freedom Caucus “sympathizer,” though he is not a member, said he expected a similar dynamic this time.

“Remember Paul Ryan didn’t become speaker because he ran for the job,” Mr. Gaetz said. “In our conference, running for speaker is a lot like running for pope. It’s not something where I expect vigorous zero-sum style campaigning.”

Still, many Republicans say that it is far from clear if a unifying figure will emerge — and that the dynamics will be different depending on whether the race is for speaker or for Republican leader. If the race is for speaker, which requires the votes of a majority of the entire House, the candidate will need the backing of the Freedom Caucus, which has about three dozen members.

But a race for majority leader, which simply requires the votes of a majority of the Republican conference, can be won without the Freedom Caucus’s support.

Representative Tom Rooney, Republican of Florida, who is retiring, said that the current situation, in which the Freedom Caucus “can basically sink any bill at any given time,” was untenable, and that he hoped the next leader of the Republican conference would be able to put an end to it.