WASHINGTON — Two months after the Pentagon began requiring all troops to get the coronavirus vaccine or face dismissal, the vast majority have now had shots, in part because none received a religious exemption, military officials said.

While vaccine exemptions are often broadly worded, requests based on religious beliefs are coming under close scrutiny in the military and at the Department of Veterans Affairs, the first federal agency to impose a mandate. They will likely be followed by the rest of the federal government, where most workers are required to be vaccinated by the end of this month. The Biden administration will release a federal vaccine requirement for private companies with 100 or more employees “in coming days,” a representative for the Department of Labor said this week.

The widespread federal and private sector mandates pose a test for the country, and the military and Veterans Affairs are being closely watched by companies and legal experts. Across the country, there are at least 40 legal challenges to vaccine and testing mandates issued by cities, hospitals, universities and other employers that have yet to move forward, while others have been knocked back.

A federal appellate court removed a temporary injunction last week that had allowed health care workers in New York to seek religious exemptions to the state’s mandate.

The Defense Department has granted a smattering of exemptions, including to people who were already leaving the military or have medical issues. However, some of those exemptions, like for people who recently had the coronavirus, may soon be reversed. Officials declined to say how many service members had requested an exemption but said the number was not large.

“I don’t see the courts interfering with the vaccines in any other context than possibly religious exemptions,” said Dorit Rubinstein Reiss, a law professor and expert on vaccine mandates at the University of California, San Francisco. “But I don’t know if courts will be willing to second-guess the military.”

The leaders of most major religious organizations have recommended that their members get the vaccine. Officials say that no one is actively discouraging people in the military from seeking a religious exemption. But anyone seeking one from the Pentagon or Department of Veterans Affairs would be required to have an established history of adherence to a religion that prohibits vaccines, among other things.

“If members of the military want to apply for one, then they should be able to,” said John Kirby, a spokesman for the Pentagon. “And they should be able to make their case.”

About 97 percent of the country’s 1.3 million active-duty service members have had at least one dose of the vaccine, and roughly 87 percent have had both shots. The Air Force, which this week became the first division to hit the deadline for its mandate for full vaccination, will release its latest vaccine data on Wednesday. About 11,000 of its 326,855 active-duty personnel are likely still unvaccinated and facing possible expulsion.

At the Department of Veterans Affairs, where thousands of workers who interact with patients were supposed to be fully vaccinated by Oct. 8, officials have taken a dim view of such exemptions. Since the department issued a vaccine mandate for its 115,000 frontline health care workers this past summer, about 88 percent of the 380,000 employees covered by the mandate have had at least one dose, falling short of officials’ hopes for nearly full vaccination.

In hospitals or nursing homes with particularly vulnerable populations, “I think that there will be a point there where it is an undue burden on us to ensure safety in the provision of health care,” said Denis McDonough, the secretary of the department, “at which point we’re going to have to deny religious exceptions.”

Vaccine reluctance in the military and the Department of Veterans Affairs mirrors that of civilian society, where vaccine rates are largely lower without such mandates. Some people have embraced vaccine conspiracy theories or have been fearful of possible side effects, or do not see themselves at risk for the virus.

Over the summer, as the Delta variant surged, military officials became alarmed at the growing number of deaths; more active duty members died from the virus in the fall than in all of 2020, and none of them were vaccinated. In total, 71 service members have died. The Navy has led the charge with vaccinations, and roughly 99 percent in the service have had at least one shot ahead of the deadline to be fully vaccinated by the end of the month. Sailors were made acutely aware of the dangers of the virus early in the pandemic when an outbreak occurred on an aircraft carrier deployed to the Pacific Ocean. The captain, who pleaded with the Pentagon for help, was later fired.

There is a “common understanding that we often work in congregate settings such as shipboard environments and it is understood that in these settings, infectious diseases can spread very rapidly,” said Capt. Robert Hawkins, who leads the Navy Medicine’s Commander’s Action Group. “Immunizations have played a large role in readiness to conduct our mission for a long time, so we have had an understanding of their role in protecting our health and mission.”

In the Marines and the Army, about 93 percent of all active-duty troops have been at least partially vaccinated. Each service branch set its own deadlines and complex disciplinary procedures for those who decline shots, including extensive counseling sessions with clergy and commanders.

Still, only a doctor can give a medical exemption. “It’s a lawful order,” Mr. Kirby said of the vaccine mandate, and commanders have the right to “ultimately do what they need to do for the readiness of their unit, and if that comes to doing something of a punitive nature, they certainly have that right and that authority.”

On a military subgroup on the social news and message board site Reddit, people swapped advice on how to talk to those who were resisting a vaccine, from offering scientific evidence to refuting claims that vaccines stem from aborted fetal cells to noting that troops take far more dangerous risks in combat. Stressing health and safety and readiness is always better than threatening expulsion, commanders say.

The private sector is clearly watching. Many companies, including United Airlines, Procter & Gamble, 3M and IBM, already have mandates, and several have indicated they will allow for “limited” medical or religious exceptions. Almost a dozen states have joined a lawsuit to prevent federal mandates from going forward.