No,they do no need to have evidence. People don't always act rationally, and if you spend any time talking to a person with anti-vax beliefs you'll notice their belief isn't founded on evidence but on narrative, distrust and emotions. Which does not make them stupid, but it does make them wrong. If they had evidence, they could present it and we'd act on it, as was seen in Germany when there were concerns about a vaccine slightly increasing heart muscle inflammation risk, it was almost immediately halted and they stopped recommending it.

And I'm assuming we still have the same rule of law as now and they don't just make shit up, because I'm modeling my fellow humans as having a different (in this case, empirically and from a risk calculus wrong) perspective, not as evil. So they can't abandon democratic principles. Vaccine mandates are coming as a last resort now due to exactly those principles