an amusing but still concerning story of anti-vaxxers getting vaccinated.

What would Hippocrates say?

agreeing that doctors shouldn't be lying to the patients, even if the doctors are actually partly doing the right thing by not faking the vaccination the way those corrupt patients wanted.

The doctors shouldn't be lying to patients, but the patients are asking the doctors to lie about having administered a COVID vaccine... One lie is unethical, the other is criminal (as I understand it). So, I would say that the only "ethical" thing would be to say to the patients, "I can't administer a fake vaccine, and claim it's real". So I don't know.