This is not simply about freedom/autonomy ‒ that cancels out. One person's right to not have the vaccine put into their body, is exactly counterbalanced by another person's right to not have the virus put into their body by that first person.
In most cases there will be at least some reduction of transmission from vaccination. But even if not, there will always be some other effect on other people: if you refuse vaccination, your illness will tax others in supplying your medical treatment. A reply will then have to move to comparing the kinds of effects/constraints on people: providing medical treatment is not as demanding as accepting an injection. Often deployed here is the term ‘bodily autonomy’, but this is misleading, since we have now left behind ‘freedom’, and are weighing other things. And then what is so bad about an injection? It seems no more than a pre-scientific superstitious feeling, and a spurious special concern ‒ since we are all anyway unquestioningly eating and breathing in the other products of our artificial societal machine.
In interesting/hard ethical problems your rejection of (bodily) constraints will impose (bodily) constraints on others. This is a general truth: the individual freedom perspective can never answer hard ethical questions, because they are about conflicts of freedom.
Instead, we should change our view and see this as a multi-agent system that we want to optimise, in this case for limiting virus flow and effect across the population. And we do so incrementally.
The justification for an ethical proposition framed this way will be by the overall good aimed and optimised for. Which has an easy form here: we do not have to surmise some likely consensus, we simply look back to the things we would normally do. The justification of any rule is found in showing how it follows from, is in concert with, what is already believed. We already have examples of what is good, and we will also have ways of judging similarity in those particular areas. So we can always take some inferential step to extend our ethical models. And this is the second general truth: ethical justification must be incremental since our plans must always work from what we have. In this vaccination case we need not argue for an expansion, but merely a retrieval of our normal activities. So unless you want to argue that our whole system/society is unjustified, you have to accept that this particular optimisation to recover it is justified too.
We do this by new rules, derived from that global aim, that slot into the place of the original rights conflict: in each interaction, we must accept some constraints on spreading, and some risk of infection.
But those rules will inevitably be imprecise, and that addresses the original concern of freedom.
A rational conception of ethics is grounded on real world effects, and directs us by generalised models of how those can be brought about. So decisions about ethical constraints are bounded by our knowledge: ethical reasoning fundamentally partitions problems into parts we do understand and parts we do not. And here we find a concession to the first complaint. We can guide purposive and coherent action, but at the same time make room for some individual choice. People have a choice at each of those situations, but those smaller choices do not matter to the overall plan.
And this is the third general truth. The problem of ethical reasoning is at its core decided by teasing apart what is known from what is not known.
Yes, some imperative of vaccination is justifiable. Because the freedom objection cancels out, the overall aim is already accepted, and some leeway for freedom remains anyway.
This, to be clear, is a somewhat different point to advocacy of strict enforcement. If you achieved 70% population vaccination, despite no explicit mandate, the imperative is substantially fulfilled. Is that dodging or finessing the question? The project is a systemic one, to attain goals at that scale. This does not necessarily require certainty or uniformity at the detail level.
Freedom's gist commonly impels such disputes, but what it offers is superficial. It is composed only of sentiment, of want, and so its evaluation is simply assent ‒ individually subjectively decided. That is, it is not really determinate, it has strength of feeling but no sophistication behind it. That is a poor principle. Instead, if we replace freedom with an incrementalising of aims and sorting by knowledge, we can make a usable tool for breaking down ethical problems. The benefit is in gaining a software-like structure: the fine granularity allows more (intricate) patterns to be matched, and the articulable modularity allows the parts to be recombined and reused. The fact that this does not yield a simple answer is what is good about it.