I don't know. I got an ID way back when I was an idiot teenager. It wasn't very difficult to figure out nor expensive. As far as Illegals go, everyone needs a vaccine no matter their immigration status. I'd like to have the ability to make sure they get vaccinated while also stopping the flow of unnecessary potential contagion coming across the border into the country every day unregulated.
See, that's the crazy part. My government is like "Everyone can get a shot, regardless of immigration status", and yet they require you to show ID before you can get one.
So yeah, maybe the homeless and (non-documented) immigrants could be super-spreaders, but it's sure isn't their fault. Heck, I feel that you won't find many anti-vaxxers among them, so it really annoys me that we're spending all this effort on giving the vaccine to those who don't want it, but still keeping it away from those who do.
I get that there is a concern that you want to keep track of the doses and ensure people aren't getting triple vaccinated, etc, but there has to be a better way.