When I helped my parents cut the cord 5 years ago, this was one of the things that I couldn't find a solution for. My father liked watching Fox News, but there wasn't a good solution to replace it other than paying for a TV Service.
There is a Fox News app, but IIRC, it is only clips of things. You had to sign in using a TV service to watch live content and full shows. I think it had commercials, too.
YouTube has a lot of Fox News content for free (YT ads, of course), so that might be an alternative. I prefer independent news content, and hardly ever watch Fox or any of the big news channels, but I do occasionally watch the Greg Gutfeld show on YouTube. Probably not the full episode, but they usually post the main story and a second clip every night around 11pm eastern time.
After installation he told me about how he used to watch Fox News for free (either through Roku or Fire TV) but could not remember how it was set up.
I am thinking that it was probably not a legit way.
I am simply baffled by the fact that cable networks that do just as much advertising as over-the-air networks do, require an ongoing cable or streaming TV subscription.
I am thinking that eventually Fox News will have paid content available on the app at some point, but I suspect that they get a decent % of TV service subscriptions that is guaranteed money, and when negotiating this with the TV service companies, if Fox News offered a way to consume the content other than the TV services, they would probably not get as good as a deal.