You can force a computer to manage two USB microphones badly, but not three. Your description of the show practically demands a small sound mixer.
Before I dig in, please know that if you go this way, you will record a Mixed Show. All the voices on one sound file. There is no going back later to spruce up one of the three voices. Audacity can’t split a mixed performance into the original performers.
This is how I do it. That’s a small, four-microphone sound mixer, but it’s entirely analog. I use a small Analog/USB adapter to get into the computer. That’s the little box in the middle. On a Mac, I don’t need the adapter because my Mac has the digital adapter built-in.
The mixer is about $120 USD. I’ve shot some very good stuff that way. This is a broadcast radio show I shot. The microphone is different, but everything else is the same.
In fact, you can get an entirely self-contained USB mixer. I think Behringer makes nice ones. I have no hands-on with those. No judgement call. I just don’t know. I would personally stay away from “advanced” mixers that feature built-in special effects, filters and tools. Those are impossible to troubleshoot and once you burn a filter or effect into a voice, it’s usually permanent.
That’s describing a straight recording featuring multi-microphone, similar to, say, a TV news show. Turn his microphone off. Turn her microphone on. Turn them all on for “banter” between news segments.
You can get crazy. People make microphone mixers that present each microphone on is own isolated channel to Audacity. In that case you can repair or filter one microphone or voice isolated from all the others and mix down later into the finished show. Multi-track connections are notoriously difficult, and not all multi-track mixers work. You’re into Unicorn Audio territory.
Audacity stands ready to save a multi-track recording for production later. An Audacity Project will save many separate sound tracks just as you see them during the recording. But nobody else can play those. You still have to Export a real stereo or mono sound file for publication.
https://forum.audacityteam.org/t/multi-channel-recording-in-audacity/15644/1
Let us know.
Koz