“By shutting down the government, Republicans are satisfying the Koch brothers while millions of people are suffering,” Senate Majority Leader Harry Reid, Democrat of Nevada, said in a Twitter message before the Kochs distanced themselves from the Heritage Action effort.

Mr. Holler said that one issue dividing conservatives was the timing of attacks on the health care plan. The supporters of Heritage Action’s strategy believed that the critical moment to mobilize was the convergence of the new budget year that began Oct. 1 — meaning the government would run out of money if a new budget was not passed — and the opening day of the online markets, known as exchanges, that enabled people to buy health insurance under the new law.

“If there is a better strategy than defunding, we are all ears,” Mr. Holler said, recalling a conversation among conservative activists this year as they debated the best path. “If it is more workable, sign us up. But nobody was able to present one that would work before October 1st.”

Opponents of the approach are arguing that conservatives would have been better served by trying to force an overall reduction in federal spending and tying that effort to the debt ceiling fight, a step the Koch-backed Americans for Prosperity urged this week. Specifically, the group argues that stopping the health care plan should be a years-long effort to marshal what it says will be growing public animosity toward the law. That will help elect Republican majorities in both houses of Congress, which will repeal it.

Stanley S. Hubbard, a Minnesota-based television executive and a donor to groups supported by the Kochs, said he preferred that approach. “Whether you like it or not, it’s the law,” he said. “And if you don’t like it, elect people who can repeal it.”

The start-up of the program offered a perfect opportunity to advance that longer-term approach, supporters of it said, since without the politically manufactured budget crisis, there would have been more media attention on the program’s troubled beginning, including failures of the online enrollment system.

“We believe tying the fight over Obamacare to the continuing resolution takes our focus off the many flaws of Obamacare, as well as cutting out-of-control government spending,” said James Davis, a spokesman for Freedom Partners.