Well the government has handled logistics related to covid horribly so far. They were completely unable to distribute tests, even months after there were tens of millions stockpiled. They were similarly bad at distributing PPE. They couldn't even get toilet paper distributed properly. Of course there are major differences here, but I don't have a lot of faith.
Government != military. The military has its own logistics command that, like the parent comment mentions, is unbelievably well tuned. Vaccines are something they do extremely well-- every sailor, soldier and airman in the entire 5 arm military needs regular vaccinations, even when deployed. Can't have GI Joe catching a flu, giving to the rest of the battalion only to sitout the rest of the conflict. All the stuff you need to safely distribute a vaccine, they've got, and know how to use it. Things like a chain of custody to keep cold things cold, and SLAs to meet perishable goods requirements, transport and delivery from remote locations with little infrastructure, the command and management to ensure enough supply gets to where it needs to be, etc. From what my friends who served in the Navy & Air Force said, they don't fuck around with vaccines. You get lined up, jabbed or misted with the vaccine and sent on your way. There's no reason they can't scale up from hundreds of thousands of soldiers to millions of civilians.
Ah, yes, I remember the days of pneumatic vaccines. Can get a lot of people vaccinated in a hurry that way.