The pinned thread in this very forum is a good place to start.

As for Chrome OS, broadcasting software like OBS should be able to run as it’s multi-platform, but you’ll need to research the drivers and supported OSes for whatever capture card you go with. I’ve never used Chrome OS so I can’t speak for the compatibility of it or what cards it works best with.

I found this relevant thread on running OBS on Chrome OS: the conclusion is that it may be possible to run OBS via Linux mode, but without guarantee. The thread also mentions Lightstream, which works via browser but is a paid service.

The most commonly video capture device for composite video is the GV-USB2, which costs around $30-$40.

Don't forget, there is always the option to just point a camera at the screen. It won't be fancy, but it will usually be allowed.

Just want to point out that last I saw (which was years ago now) gv-usb2 only had Windows drivers, so I’m not sure if it would work with Chrome OS or any Linux OS for that matter.