Federal officials have said HealthCare.gov was overwhelmed by the volume of visitors right after it launched on Oct. 1, an admission that has drawn harsh criticism from tech experts. They note the many online shopping sites such as Amazon.com that routinely handle significantly higher volumes of visitors each day.
Lauer also said that "a little bit more than half" of the 20 million visitors to ehealthinsurance were "between 18 and 34 years old."
That demographic is a crucial one for Obamacare insurance plans, which by law cannot bar covering people with pre-existing conditions. If not enough young, healthy people sign up for those plans, insurers would likely be forced to increase their premiums in future years to cover the benefits they have to pay out to older, sicker people who sign up.
Younger adults "are not going to go to the call center or use paper applications," said Lauer, referring to the alternative routes that federal officials have suggested people use while HealthCare.gov's online enrollment system continues being fixed. "They're going to do this online—period."
Lauer concedes that his idea of taking over the shopping and enrollment application of HealthCare.gov is a long-shot.
The Health and Human Services Department, which operates HealthCare.gov, is not likely to take eHealth up on its offer because officials there have repeatedly said since last week that HealthCare.gov is working better every day with the fixes it is making, and the agency expects the site will be working smoothly for the vast majority of users by the end of November.
Still, the technical feasibility of Lauer's proposal is not far-fetched, according to other Web insurance markets, and insurance experts.
Last summer, the federal government signed a deal with 34 web-based markets, including eHealth, GetInsured, and GoHealthInsurance, among others, to allow them to sell the same insurance plans that are being sold on HealthCare.gov to individuals who would receive government subsidies to offset the cost of that insurance.
The federal government cut that deal, to exploit the marketing muscle of sites like ehealthinsurance, which consistently is a top result on Google searches for the words "health insurance."
But none of those companies has been able to connect with the federal data hub in HealthCare.gov to begin selling those plans because of the same tech issues that have hampered other parts of the federal marketplace.
(Read more: What did Obamacare czar Sebelius know about tech glitch mess?)