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Mathjax is a JS program that runs on websites and renders fenced Latex code.

If you just want to see how it renders, there are countless web repls out there. https://www.mathjax.org/#demo is one, this site is another. From your question, it sounds like you are trying to debug a complex Mathjax piece and the web repls don't give you enough detail about what's broken. Most people just use some Latex compiler, and then trust that "if it works in Latex it should work in Mathjax" - this has the same environment issue though.

You can run Mathjax locally with nodejs, that is probably the most direct way. Their own docs explain it: https://docs.mathjax.org/en/latest/web/hosting.html But if your problem is predicting what Latex packages are supported on some 3rd party site (like Codidact) this probably doesn't immediately help as you would need to know the details of what they provide.

To be more precise, it seems like Mathjax doesn't handle Latex packages because it doesn't implement full Latex. Instead they implement some Latex in JS, and provide a limited interface to extend it. See https://docs.mathjax.org/en/latest/input/tex/macros.html#tex-macros and https://docs.mathjax.org/en/latest/input/tex/extensions.html. So if your issue is matching the environment of some website, the only surefire way is to figure out specifically how _their_ Mathjax is set up and replicating on your side with Mathjax.

That said, and while I'm also curious now if there's a way to do it, Mathjax seems like something where you just keep it simple and hope you get lucky and it just works. If you have some complex Latex, rather than fighting Mathjax, you could just convert it to a PNG or PDF (with a real latex tool) and put that online.