Those people who do not trust the government are not the target audience for such actions. They are trying to lure those who are hesitant or just lazy to get vaccinated. Based on the example of my country, I can say that the effect of such actions is about zero, and maybe even negative - all people understand that free cheese is only in a mousetrap.
A little carrot and stick method won't hurt, I mean if it does the job, we have no questions about it right and I think that it's effective since it's making people encouraged to get vaccinated and given how a lot of people loves lottery, who would refuse a free ticket just for some doses.
Given the huge number of side effects from this vaccine, plus the fact that these vaccines have not yet stood the test of time and the full cycle of safety checks, this lottery looks immoral to me. This is a risky decision and I don't think there should be any incentives like gingerbread. And I don't like the idea of wasting taxpayers' money on such nonsense.