What about someone who refuses because they are against the vaccine (the anti-vaxers)?

It is a theoretical possibility that “anti-vax” beliefs could be a philosophical belief under the Equality Act 2010 and therefore anti-vaxers have the right not to be discriminated against for their beliefs. Much will depend on why the individual is against the vaccine. Conspiracy theorists (the vaccine is being used as an opportunity to monitor you or it’s all because of 5G) are highly unlikely to be treated as having a philosophical belief!