Take Live Photos
FaceTime allows for quick GIF-like Live Photos that will capture a short section of your meeting.
Check out the shutter icon at the bottom of the app while on a video call, and click it. This will immediately screenshot your current FaceTime screen, and save it to your Photos app. You should get a notification when this process is complete.
Note that the Live Photos option in FaceTime settings needs to be on for this to work, and it may not work if the other person doesn’t also have Live Photos enabled.