That's part of the logistical challenge of this Pfizer vaccine that I'm trying to wrap my head around. You have hospitals nearing capacity, banning visitation, but also they're the only ones with the ultra-cold temp infrastructure to store the vaccines for hundreds of thousands of people in their community? Who are the people on the ground actually administering the vaccine? In some states they're calling in health care workers who are currently out with COVID because they're so understaffed. Does every community have some untapped pool of nurses that can do nothing but administer vaccines 24/7 for 2-3 months (or more)?
Each hospital/clinic/community is going to have an individualized plan. It will absolutely be a challenge for all the reasons you mention. We are going to have widening health disparity issues in the distribution of a vaccine in this country.
The hospital may be the site of storage and distribution, but it isn't going to be hospital nursing staff administering vaccine. And likely it will be drive up so people don't need to go into the building. In our clinics it is primarily medical assistants and LPNs who administer vaccines. My health system is integrated hoslital/clinic. They will likely pull MA/LPN staff from clinic sites to administer vaccines and hospital RNs will continue to do higher skilled care in the inpatient setting. There is still a staff shortage in the clinic setting due to illness and quarantine, but the clinic is able to absorb shortages better than the hospital.
Right now we are in flu vaccine distribution season. It's a very different work flow from prior years. There are 4 clinic drive up flu sites by appointment every Saturday that can each handle 300-400 per day (and we are one of 3 health systems/hospitals in the community, each doing something similar). It's staffed by MAs and LPNs from the clinic. I give the administration credit, they pay overtime for the flu clinics and staff have signed up voluntarily for those extra shifts. Even with a certain percentage of staff out sick we can manage that staffing. It will likely be a similar model for COVID vaccine distribution. But not as high volume. Instead of distribution over 2 months, COVID vaccine distribution is going to be in phases, starting with highest risk, over 4-6 months. There are a lot of people who are not interested in getting the vaccine. I don't know that the initial acceptance rate will be as high as we hope.
For those without insurance or a primary care "medical home" the health department will have a certain number of free doses. They may do distribution from trucks in a parking lot, probably not the pfizer version because of storage issues. Although hypothetically if Pfizer could partner with health departments and allow distribution directly from the transport trucks (my understanding is that Pfizer is doing their own distribution, although early on UPS was investing in cold storage and trucks as well) it might be possible. But I'm not sure they will have the supply to move into that part of the market early on.
Unfortunately once we get everyone vaccinated who would like it over the first 4-6 months it will likely be time to start over with boosters for the initial group.
ETA: I forgot to clarify, the nursing staff called back to work when COVID positive are only those who are asymptomatic and working only with COVID positive patients. Not that I'm saying this is a good thing, but it's very limited in scope in very few hospitals across the country.