Pulseaudio has network support. You should be able to share your microphone. Of course you need to run pulseaudio inside the container, too. So it's no longer an application container, but a system container.

I used system containers in the past, they are not always easy. No idea whether it would work with podman.

Now that I think about it: The browser obviously uses the display of the host. How? Only over X11 network protocol? That worked in the past, does it still work today? I thought modern browsers would need /dev/dri/? If that is made available, why would Webcams not be possible?