WASHINGTON — First there were culture wars over lockdowns, then over face masks, then over face masks in schools. And even as the pandemic appears to be receding, a culture war over vaccine mandates is showing signs of only growing more intense, with Republican governors and legislators launching challenges to rules that, many public health experts say, could help ensure there is not another coronavirus surge in the future.

Resistance mounted as soon as President Biden rolled out the new mandate on Sept. 9. The mandate, which targets any business with more than 100 employees, is expected to cover some 80 million workers across the country. It won’t go into