I wouldn't say that someone has a right to know the health status of individual coworkers. Although they have a right to ask, as long as it's in a conversation where all participants are happy to talk about it, it's not causing animosity and they're not harassing anyone over it (and others have a right to decline to answer).
If he randomly goes around asking "Have you been vaccinated? Have you been vaccinated? Have you been vaccinated?", then you should tell him to stop that. If he wants to discuss this in a respectful consensual private conversations with people, he can do so, but randomly asking people that out of the blue would be crossing the line of what's appropriate workplace conduct. If he gets to a point of harassing people about it in public or private, then just treat that as any other form of harassment.
The above said, I would say they have the right to know whether some environment they frequent is protected from a deadly disease through widespread vaccination and/or testing.
I might even go one step further and say they have a right to work in such a protected environment, because no person should have to choose between their job and their health.
Although, even if you think COVID is practically harmless, I would still argue that they have the right to know whether an environment is protected so they can make an informed decision about their health. If they want to worry about COVID, it's their right to do so.
At least be upfront and clear with them about whether you're going to enforce vaccination or testing in any way, now or in the future, so they can decide whether to start looking for a new job, or to take any other steps they deem necessary to protect their health.
Note that the above is based on workplace etiquette and morality, not on what the law says.
Local laws may require employers to enforce vaccination or testing of employees, so do make sure to know what those laws are and to follow them.