Having discovered this 15 minutes ago I can't really speak in absolutes, but firing up OBS and using their test server, I think this would allow you to do exactly what you want. It's trivial to get OBS set up to stream to this, and while I haven't tried it, the docs say that it supports multiple streams to the same stream key. I guess it would just show each stream side by side in the client?

I think you could run your own server, run an instance of the front end and distribute the instructions to setup OBS to your family.