R(eff) lowers the higher the vaccination rate. There are some pretty obvious % targets depending on your R(eff) initial. For the initial population number it's ~90%.[1]
"no reliable evidence" - of course vaccination helps bring R(eff) lower. That's just the way the math works. If R(eff) is already low enough due to other mitigation, then additional vaccination helps bring it below 1. We see that all over the nation now, R(eff) rates dipping below 1 as vaccination rates increase.
https://www.businessinsider.com/delta-variant-made-herd-immu...