Stream ripping software ?

There's most certainly rippers for regular sites like youtube, etc. This would be yt-dlp which you can google for a list of supported sites/services that is changing/updating all the time. For Amazon and others like that they usually require your cookies to authenticate the fragment files linked-to / listed-by the m3u8/json playlist file for the video. Another big problem is even if you can grab the fragments (or defragment to a full video file), often time they are encoded via WideVine/FiarPlay or other DRM techniques. I haven't looked into many of them but I know the DVDFab products seem to offer ripping ability for a variety of services. I tend to not use paid products or freeware but instead look for open source things to use which are free but without all the useless ads and crypto mining in the background.

The next best bet is a screen recording software to take a recording of the whole show/movie/whatnot or even a hardware setup that you can have as a 2nd screen for your pc or hdmi from a tv or something and have it record the input to a file instead of actually displaying it. Lots of research to do though, I haven't used either of those for this purpose so I'm not sure what exactly would work best.
If you plan on screen recording and can afford it, a VPS could help you to stream and record without having to use your own machine so you don't accidentally start doing something else while it's recording, plus storage/bandwidth would likely be higher.

I'm not on here often so if you reply I likely won't be able to get back to you for a while.