Got to play with it a bit over lunch. Turns out the desktop application is no good for linux, but I was able to use the Firefox and Chrome extensions to use Webex through the browser. For anyone who comes here looking for help you'll want to do this too.
Don't try to 'install' WebEx. It won't work. (I qualify this statement that as of March 2020 it doesn't work.)
Instead go to the web store for your browser of choice. Search for the Cisco Webex Extension and install that. You may need the 'Cisco Webex Content Sharing' if you plan to share your desktop, but I haven't tested this. After installing don't try to join the test meeting that webex hosts - it wont work. Instead you have to go to webex.com and click 'Join' there. Give it the meeting number (and your name and email), and it should connect you. I was able to connect to my private webex room (my workplace uses webex) without issues. I tried it in both Vivaldi and Firefox, and both worked with the built-in webcam and microphone.
Best of luck!