Firefox's VPN, says "Check your Internet connection" even though I am connected to the internet.

Are you enrolled in the Beta deployment already, and are in the eligible countries, or you had to flip the switches yourself manually to enable the feature?

I didn't have to do any enrollment, it just showed up as a part of my installation of 152.0. (I am in the United States.)

There's a lot of endpoints the service needs to reach, do you use any filtering DNS or a networking setup (e.g. pihole) that would limit some datacenter traffic? Or ports restricted? An example endpoint is https://us.m1.fastly-masque.net:2499 — can you even connect to that? (it shows "method not allowed" in a browser, that's correct result)

I get "method not allowed" when trying to reach that endpoint. Not using any filtering DNS in my setup. I do wonder if this was just some strange transient issue, because now when I click the VPN button it is working fine.

That looks like you should be able to connect just fine at this moment.

In case you see rather incorrect connection errors in the future, feel free to tinker with DNS–over–HTTPS settings and enable/disable or change the endpoints used — it might "place" you in a different geography and hook you up to a different datacenter in case the one picked was not reachable for you for some reason.