Yeah it's going to be a challenge, but you could distribute it the same way you do testing with drive-up/walk-up using air injectors like they use in the military. It's far simpler than a nasal swab and we already test ~1M/day. If it's part of a workplace requirement (first responders), you're likely to do everyone over a week at work anyway. It's unlikely that even in developed countries the size of the US with multiple vaccines there is going to be more than 50M doses/mo and one should expect it to take 6-12 months to initially distribute the vaccines. Note that it's only ~1.5M/day and that a single (2-4 lane) drive through station open 10 hours could pretty easily do 1000/day. I'd hope there are at least 1500 stations to cover the US to ease transportation difficulties (roughly one per 30 zip codes). The idea that the entire US couldn't reliably mail out samples on dry ice (enough to last ~3 days) to 1500 locations a day speaks more to the sad current state rather than what is possible in a normally functional mail system.
Bikeshedding on individual vials and syringes for less developed countries borders on silly since most other vaccines aren't distributed that way. Vaccinating the 4B people in less developed countries is going to be harder, but it's doable.
The real concern should be whether enough people will choose take the vaccine and whether vaccine (or post infection) protection will be durable (or neutralizing).