Distribution in the country is not the problem (look what "Medecins sans frontires" do). Mostly 3rd world countries do not have medical standards as high as in first world so you can instruct taxi drivers how to jab people and give them trunk full of vaccines and they will empty it in the same day without problems. And it will be still safe enough as those vaccines are applied to muscles.

I think it is the problem of supply and demand because EU and even more US* are making vaccines way more expensive.

* EU is buying vaccines together to not compete for price between states but the price is probably way higher than African countries can pay. Imagine the whole world teamed up together and vulnerable people would be vaccinated at the same time before young healthy people as it is done now in EU.