There's a correlation between religiosity and anti-vaccine sentiment, but I'd go as far as to say we have good reason to think there isn't causation. Case in point: the LDS church leadership is insisting on vaccination and masks during services, however individual Mormons are fiercely opposed to these positions [1].

I think a better explanation for the correlation is that those who derive their identity from the white evangelical subculture feel that their own church leadership don't best represent what it means to be a part of that group. I assume thought leaders like Tucker Carlson and Donald Trump hold more legitimacy in certain respects with evangelicals than explicitly religious leaders do. I don't think labels like "evangelical", "white", and "conservative" properly capture this group; it's more of a synthesis of these individual identities that have mobilized into a greater reactionary movement.

[1] https://apnews.com/article/lifestyle-health-religion-coronav...