If you were using an aerosol virus it would have to be a live attenuated vaccine. People might be inside or at sea or wearing a mask or otherwise might not inhale the vaccine during the period when it was airborne and you would have to have these people catch the vaccine from people who were vaccinated.
There are vaccines that work this way. The Sabin polio vaccine is a great example.
An important feature of Sabin’s oral polio vaccine was that
immediately after vaccination, people shed weakened virus in their
fecal waste. This boosted immunity for others in the community and
gradually reduced the number of people susceptible to poliomyelitis.
Between 1963 and 1999, Sabin live vaccine largely replaced Salk killed
vaccine everywhere in the world. However, because the live virus in
the vaccine occasionally became strong enough to cause actual disease,
Salk killed-type vaccine has replaced the live type in the United
In places where there is potentially real polio, the benefit of having vaccine transmitted person to person and prevent polio outweighs the occasional case of polio caused buy the vaccine. In the US where there is little polio (they graciously do not mention that usually in the US, our drinking water does not contain the poop of our neighbors so there is less transmission) the risk of vaccination induced polio outweighs the protection from real polio conferred.
In your situation the mother colony probably does not care too much about sickly people with weak immune systems. A live organism vaccine which was very contagious would be optimal for rapidly conferring immunity to the populace and a few cases of the actual disease among nonworkers would be acceptable. If you want them to seem like Hunger Games type hardasses you could make the vaccine itself be pretty tough on the populace - maybe scarring them up like smallpox used to do.
But back to real life and the WHO: generally for real vaccines the live attenuated vaccine is actually the pathogen itself and a weak strain is selected. Recombinant epitopes are already used in the hepatitis B vaccine. A transmissible norovirus (for example) engineered to express polio antigenic epitopes would have no possibility of causing real polio, but could induce immunity to those epitopes. Hopefully real vaccines are made that way in the near future because that would be much safer. A benign and paternalistic mother colony (can you have a paternalistic mother?) could cause everyone in the colony to get "stomach flu" or some even less symptomatic disease with a very contagious vector that was engineered to express the relevant epitopes.
Infrastructure: You could spray an aerosol rhinovirus out of planes like crop dusting, or from trucks like they spray for mosquitoes.
Respray: it depends on the disease. Flu vaccine is every year because there are antigenic variants that come up. Polio vaccine lasts for life and so you would need to respray when there was a new generation of kids born since last spray.