This sounds like a good idea on paper, but allow me to frame this differently. The first vaccine will likely be slightly experimental because it was done rather quickly. This means you are effectively asking the government to set up shop in likely minority and poorer neighborhoods to administer a vaccine that might have unknown side-effects that don't appear until being used by a larger population of people. It seems like you are supporting experimenting on the poor.

I think you need to make it available, but in this case, maybe allowing people to choose is the right decision. Make sure it's available equally, yes, make sure poorer people have access, but don't rush to make sure that only the poor get it.