I can't speak to every hospital's vax mandates, but ours does not. I'm not aware of any small town hospitals that do.
Areas to avoid: the reservations are obvious. Bottineau isn't on the res, but it's too close for comfort. There are areas of Fargo that are best avoided, mainly the northwest quadrant of the city if you use the interstates as axes. The northeast is also infested with college kids and other low income housing. Move there at your own risk.
If you're considering rural life, people east of the Missouri river tend to be more neighborly than people to the west. Ranchers in the west are very reluctant to welcome outsiders.
If you do move here, do your best to assimilate. There's a lot of gossip in small towns. Please, please never talk about how "we did it back west." We're a friendly people. But we like our own ways.