If the immunity lasts for 4 months, we are talking about yearly vaccinating 200% of the population.
It's not impossible for countries with a small population, but AFAIK the flu is the largest scale vaccination in the world, and we are talking about something 4 times larger (than those countries that vaccinate the most - how are China and India numbers?).
But then, this seems to be the worst case with a reasonable likelihood. Things will probably be better.
I think we will be more limited by supply rather than willingness. Locking down for months has much more economic impact than one shot for each citizen even for poor country.
[1] https://www.canada.ca/en/public-health/services/publications...