Governors from five states have written a joint letter to Lloyd J. Austin III, the defense secretary, asking that their National Guard troops be exempted from a federal coronavirus vaccine mandate, greatly escalating what had been a single state conflict over inoculations.

“Setting punishment requirements for refusing to be Covid-19 vaccinated, and requiring separation from each state National Guard if unvaccinated are beyond your constitutional and statutory authority,” wrote the Republican governors of Alaska, Wyoming, Iowa, Mississippi and Nebraska to Mr. Austin, and asked that their states be given an exemption from the requirement. The Pentagon has yet to respond to the letter, which was dated Tuesday.

Last month, the Defense Department rejected an attempt by Kevin Stitt, the governor of Oklahoma, to exclude the state’s National Guard from the mandate, setting the stage for Guard members in the state to lose their jobs should they refuse. The department this week released a new order noting that all guard members who do not get vaccinated will receive a general order of reprimand, which would essentially end their career.

Federal officials have said that governors have no legal standing to allow Guard members to refuse the vaccine mandate. State officials and some legal experts, however, believe that unless National Guard members are federally deployed, they are under the jurisdiction of the governor of their state and therefore not subject to federal mandates.

Since August, when the Pentagon began requiring all troops to get a coronavirus vaccine or face dismissal, roughly 97 percent of the 1.3 million active-duty service members in the United States have had at least one dose of the vaccine, but the numbers are much lower for members of the Guard and Reserve.

Each service branch has a different deadline for shots, and the Guard units will not face theirs for months. By increasing the number of states tangling with the department, the governors of the states that have generally resisted vaccine mandates underscore the increasing politicization of the mandates in a part of government — the military — where vaccine mandates are frequent and largely without controversy.

The Air Force, with more than 325,000 active duty members, was the first to hit the deadline; a mere 27 airmen — all with less than six years of service — are the first believed to have been fired over vaccine refusals, Pentagon officials said this week. The Army and Navy are expected to announce their numbers imminently.

The Defense Department has granted very few exemptions from the mandate, other than to people who were already leaving the military or those who have medical issues. So far, not a single service member has been granted a religious exemption.