Three approaches:
The medium difficulty cheap way
Set up one client machine, e.g. the one you’re using, to record both your microphone and the audio output of the web browser. Details very a lot. This usually means messing about close to the operating system level, because something like Audacity won’t record from two completely separate sources at once (though it will record a whole bunch of separate channels from the same source). I can give you a recipe for pulseaudio on Linux but I never got it working with straight alsa (even though it should be possible) and I have no clue about Windows or Mac.
In my experience this is the lowest quality.
The easy expensive way
Get a multichannel recorder; I use a TASCAM DR-40 (technically a DR-40X now, the old '40 stopped being able to write to SD card). Channels 1 and 2 are the built-in microphone, pointed at me; channels 3 and 4 come off the computer’s headphone output (I have a boring old analogue splitter which also feeds my actual headphones).
This is medium quality.
The hard cheap way
Get each individual player to record their own microphone. Windows used to have a useful Sound Recorder, but it’s been made useless of late, so the players I know use Audacity. Stand-alone recorders would also be possible Then they send me the recordings (encoded as flac, not the much bulkier Audacity native format) and I mix them together.
This is more work for them (everyone has to set up the recording and export the audio) and will be lost if someone’s machine crashes. And whoever edits them together has to get them all synchronised. I use a “chat” track from one of the previous two methods (which is also my backup in case things go wrong here) to verify that. But then I can denoise the tracks individually, clip out coughs and can openings, and put players into consistent stereo positions.
This is the highest quality I can get, and it’s what I do for sessions I’m planning to release. (It’s also how I put together the podcasts I’m involved with.)