WASHINGTON — The vast majority of active-duty troops in the Army and Navy are vaccinated against the coronavirus, and the small number of those still refusing shots will soon be dismissed from the military, officials said Thursday.

In the Army, 468,459 active-duty soldiers, or 98 percent, have received at least one dose of the vaccine. The Navy has inoculated 342,974 members, with roughly 1.7 percent still holding out. Among the unvaccinated, however, are some who have medical and administrative waivers or pending requests, or have already signed up for shots.

Army officials said that less than one percent of active-force members have refused the vaccination order without a pending or approved exemption. Thousands of troops requested religious exemptions, but none have been granted, officials said.

On Wednesday, the Senate passed final legislation that would prohibit the service branches from dishonorably discharging any member who refused to get the coronavirus vaccine. The measure, which was supported by Republicans in the House and the Senate, allows those who declined the vaccine to receive certain benefits after leaving the military.

Earlier this week, five governors wrote a joint letter to Defense Secretary Lloyd J. Austin III asking that their National Guard troops be exempted from a federal coronavirus vaccine mandate, escalating what had been a single state conflict over inoculations. While National Guard troops helped oversee large vaccine sites for Americans across the country last year, they have lagged behind their active-duty counterparts in getting shots.

“Thank you to the medical staff who have been supporting the pandemic response at home and to the vaccinated soldiers who put the health and welfare of their fellow soldiers and families first,” Christine Wormuth, secretary of the Army, said in a statement. “To those who continue to refuse the vaccine and are not pending a final decision on a medical or administrative exemption, I strongly encourage you to get the vaccine. If not, we will begin involuntary separation proceedings.”

Army commanders have relieved six active-duty leaders, including two battalion commanders, and issued 2,767 general officer written reprimands to soldiers for refusing the vaccination order, according to Army officials on Thursday.

Vice Adm. John B. Nowell Jr., the chief of naval personnel, said this week that the administrative separation processing for those who continue to refuse would be expedited to “maximize speed and equity in achieving a fully vaccinated force.”

Each service branch set its own deadlines for compliance with the mandate, which was issued last August. The Air Force, with more than 325,000 active-duty members, hit its deadline on Nov. 2. Pentagon officials said this week that 27 airmen — all with less than six years of service — were the first believed to have been fired over vaccine refusals. The deadlines for the Marines and the Navy were on Nov. 28. The Army’s was on Wednesday.

Among the Marines, 94 percent are now fully vaccinated and 95 percent have had at least one shot. Marines who did not receive a second dose by Nov. 14 are considered unvaccinated and will be processed for administrative separation unless they are awaiting a decision on an exemption request, said Maj. Jim Stenger, a Marine Corps spokesman.

The deadlines for National Guard troops are next year.

Five governors asked that their states — Alaska, Wyoming, Iowa, Mississippi and Nebraska — be given an exemption from the requirement. The Pentagon has yet to respond to their letter to Mr. Austin, which was dated Tuesday.

“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,” the governors said.

Last month, the Defense Department rejected an attempt by Gov. Kevin Stitt of Oklahoma to exclude the state’s National Guard members from the mandate. The department released a new order this week 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.

The tension between the states and the Pentagon underscores the increasing politicization of vaccine mandates in a part of government — the military — where many other vaccinations are administered with little controversy.

Vaccines are a standard part of the job for those who serve in the military. Beginning with boot camp, troops are required to get inoculated against roughly a dozen diseases.