There is no such feature in SpatialChat. If you want to record, you have to use a third-party app.
But there is a difficulty here - capturing the camera and microphone.
SpatialChat requires access to your camera and microphone to work correctly. And when such access is granted to another program, problems may arise.
Unfortunately, this varies from case to case, and we have not yet found the best solution to this task. We use http://www.loom.com to record our events at SpatialChat. Perhaps this solution will work for you too.