Albeit having a strong preference for Jitsi Meet, I had to get Skype working on OpenBSD to meet online with a client. Due to some strange, borderline-psychotic logic, Skype Web doesn't support audio or video calls out of the box on 'unsupported systems'. After some online searching, multiple sources recommend changing the user-agent to mimic Chrome on Windows, which unfortunately wasn't the fix. It still would have those audio/video buttons greyed out.
After some tinkering, I managed to get it working. The trick is stupid simple: use preview.web.skype.com, rather than web.skype.com.
Mind you, this can break at any given time - if Redmond decides to block the 'unsupported systems' on the preview version, you are out of luck. But right now, it does work. Verified it with some test calls.
As the case with OpenBSD, you do need to enable the microphone and webcam explicitly in order to use it. This can be achieved by setting two sysctl's (on -current and the upcoming 6.9 release):
# sysctl kern.audio.record=1
# sysctl kern.video.record=1...and then either making the $USER a member of the video group or by changing the privileges to the device: # chmod 666 /dev/video0. While these options can be set as a system-wide default, I would recommend against it - it tends to be a good practice to having recording and the webcam disabled by default.