A vaccine will be nice to have but it won't be sufficient to eradicate the virus. Some people won't be vaccinated. No vaccine is 100% effective. And there are animal reservoirs; we obviously can't vaccinate wild animals.
We don't get 100% vaccination on any disease, but we have eradicated a few and all but eradicated a bunch. We just need something approximating herd immunity levels. Those who want vaccinations will get them, the rest can slowly add to the total until society as a whole is no longer worried about coronavirus.
If we really get that level of abstention, then maybe it would be a good idea after all to give people who test positive for antibodies some kind of way to prove it.