The issue of slow vaccination rate may be because (1) the infection rate is low, (2) people know there are an "abundance" of vaccines available now and in the future. So there is no benefit to rush given (1).
To change this dynamics, the government can let people make appointment for a limited period of time, and if people do not take within the time period, there will be a significant shortage of vaccines and not only they need to wait for much longer, they will need to pay in the future. This changes the expectation.
Also, try to encourage the social responsibility of this instead of using awkward and complicated measures.
Nobody rushes to eat the lunch when free lunch is always available.