Asynchronous play is usually handled by exchanging logs. Vassal logs are actually saved games that include a step by step record and playback of logged moves.
VASSAL comes with a User’s Guide. You can access it from the Help menu within the software in addition to the previous link. Among the many topics are connected online play and asynchronous play.
If you’re seeking to play a session with several players “face-to-face”, you’ll need to use the Vassal server. Then one player (host) should set up a room on the server and the other players will need to join the room, choosing which side they will play during the game set up by the host.
This is explained in the Vassal guide and is fairly simple to do. When the game session is complete, the host should save the game so it is available for the next session.
Hope this helps (assuming it represents what you are trying to achieve).
If exchanging logs by discord, for example, I might also send an image of the board, so that players who don’t need to react to my log can quickly see what happened without needing to view the log.