You have not said how the database is being hosted or what version. I would assume that you are using FileMaker Server? If so, you might want to investigate the server logs and see what information they hold.
Sorry for the lack of information. My database is being hosted by pointinspace.com. I'm not sure what method of hosting they use or how they are hosting it. I talked with the owner of the site and he suggested "set up a script to run on database open that would log the current user, and maybe IP (I think you can fetch that with a script) to a table. Then, have another database close script that logs the logout."
Anyone know how to do this?