What I'm asking is basically, what's the exact legal position/incorporation of Project Warp Speed relative to the US government?
Incorporation is not required. Congress appropriated money for certain purposes. The president has a responsibility to spend that money in a manner consistent with the law. Operation Warp Speed is the name of the program to execute the law.
While almost always associated with business, a COO can exist independently of a corporation.
From Harvard Business Review, May 2006, Second in Command: The Misunderstood Role of the Chief Operating Officer.
One role of a COO is to lead the execution of strategies developed by the top management team. [...] Managing large, often global, enterprises sometimes requires two sets of hands; in such cases, the COO typically takes responsibility for delivering results on a day-to-day, quarter-to-quarter basis.
In this case, the "top management team" is in the White House. General Perna has been tasked with the execution of those strategies, thus chief operating officer.
That the appointment of a COO is reasonable for execution of those strategies is revealed by the framework and goals of the program.
HHS Press Release, May 15, 2020, Trump Administration Announces Framework and Leadership for 'Operation Warp Speed'.
Elements of Operation Warp Speed
Operation Warp Speed is a public-private partnership to facilitate, at an unprecedented pace, the development, manufacturing, and distribution of COVID-19 countermeasures, between components of HHS, including CDC, FDA, NIH, and the Biomedical Advanced Research and Development Authority (BARDA); the Department of Defense; private firms; and other federal agencies, including the Department of Agriculture, the Department of Energy, and the Department of Veterans Affairs. It will coordinate existing HHS-wide efforts, including the NIH's ACTIV partnership for vaccine and therapeutic development, NIH's RADx initiative for diagnostic development, and work by BARDA.
Leadership: In addition to the expertise of Dr. Slaoui and General Perna, each countermeasure area will be overseen by a highly qualified and accomplished career HHS scientist:
Vaccines: Peter Marks, M.D., Ph.D., Director of the FDA's Center for Biologics Evaluation and Research.
Therapeutics: Janet Woodcock, M.D., Director of the FDA's Center for Drug Evaluation and Research.
Diagnostics: Bruce Tromberg, Ph.D., Director of the NIH's National Institute of Biomedical Imaging and Bioengineering.
The Department of the Defense (DoD) has identified subject matter experts to lead DoD support for five areas, including:
Diagnostics: Major General Lee Payne, Assistant Director for Combat Support at the Defense Health Agency (DHA).
Therapeutics: Sean Biggerstaff, Ph.D., Acting Director for Research and Development at DHA.
Vaccines: Matt Hepburn, M.D., Joint Project Lead CBRN Defense Enabling Biotechnologies.
Production and Distribution: Stacy Cummings, Principal Deputy Assistant Secretary of Defense for Acquisition Enablers.
Security and Assistance: Andrew Kilianski, Chief Intelligence Officer for CBRN Defense.
Three focus areas: There are three core areas where the effort will accelerate the timeframe for countermeasures, including a vaccine, reaching the American public: 1) development, 2) manufacturing, and 3) distribution.
Commitment to affordability: The Trump Administration is committed to making these countermeasures affordable for the American people. As a condition of receiving support from Operation Warp Speed, companies will provide a donated allocation of countermeasures developed, including an eventual vaccine.
Financial resources: Congress has directed almost $10 billion to this effort through supplemental funding, including the CARES Act, and Congress has appropriated other flexible funding. Over $6.5 billion has been designated by Congress for countermeasure development through BARDA, along with $3 billion for NIH research.
Planned updates: Public updates on each step of Operation Warp Speed will depend on certain scientific milestones, which depend on the results of clinical trials (e.g., trials are sometimes halted before completion due to clear results).
Where does Operation Warp Speed fit in the US government (of GSEs) structure?
Operation Warp Speed appears to be wedged between HHS and DOD. Most of the funds, appropriated by Congress for "developing, manufacturing, and procuring vaccines and other medical supplies", were for HHS. Operation Warp Speed will use those funds and bypass the normal processes of HHS hoping to speed the development of vaccines for COVID-19.
Trump to name former pharma exec as vaccine czar, May 13, 2020
“Operation Warp Speed” is operating largely independently of the existing White House coronavirus task force, which is also shifting its focus toward vaccine development.
The initiative is being promoted by White House senior adviser Jared Kushner, and involves officials from the Defense Department and the Department of Health and Human Services.
Unveiling ‘Warp Speed,’ the White House’s America-first push for a coronavirus vaccine, May 12, 2020.
Conventional wisdom is that a vaccine for COVID-19 is at least 1 year away, but the organizers of a U.S. government push called Operation Warp Speed have little use for conventional wisdom. The project, vaguely described to date but likely to be formally announced by the White House in the coming days, will pick a diverse set of vaccine candidates and pour essentially limitless resources into unprecedented comparative studies in animals, fast-tracked human trials, and manufacturing. Eschewing international cooperation—and any vaccine candidates from China—it hopes to have 300 million doses by January 2021 of a proven product, reserved for Americans.
Those and other details, spelled out for Science by a government official involved with Warp Speed, have unsettled some vaccine scientists and public health experts. They’re skeptical about the timeline and hope Warp Speed will complement, rather than compete with, ongoing COVID-19 vaccine efforts, including one announced last month by the National Institutes of Health (NIH). “Duplication only leads to infighting and slowing people down,” says Nicole Lurie, former U.S. assistant secretary for preparedness and response, who advises the Coalition for Epidemic Preparedness Innovations (CEPI), a nonprofit funding and helping coordinate COVID-19 vaccine efforts. “The U.S., and others around the world, should be engaged in this competition against the virus, not against one another.”
Although Warp Speed has not ruled out any type of vaccine, it will not consider ones made in China, such as the inactivated virus vaccine recently shown to protect monkeys from the coronavirus, a first. “We can’t partner with Chinese companies,” the official says. “That’s just not going to happen.” The decision was “above my pay grade,” he adds. But the chosen vaccines could be made by a company that is not headquartered in the United States. (White House trade adviser Peter Navarro in a February memo to the Coronavirus Task Force championed a “Manhattan Project” for COVID-19 vaccines that would prioritize U.S.-based companies.) [Emboldening original]
While the DOD's role has not been widely reported, some assets will be used during vaccine development and testing, but mostly during the distribution phase.
Once a product such as a vaccine is ready, the Department of Defense's involvement will enable faster distribution and administration than would have otherwise been possible using wholly private medical infrastructure.