A solution could be: Have a "central location" for the database. Copy the database to the laptop before use each night. After use each night, copy the database back to the central location. Done. A simple way of synchronizing. The issue is you have two laptops running tournaments on Wed. You can copy the database from the central server to each laptop before use on Wed, but after use copying the database back from the laptops to the central location presents a problem. Any changes made to the database on the laptop that copies back to the central location *first* on Wed night will be lost when the second laptop copies its database back to the central location.
In reality, use on the laptops probably consists only of (a) adding new players and (b) changing existing players data. It's unlikely a player would be deleted from the database on the laptop, or that a change to the same player record would occur on two laptops on the same night, since a player can't be at both locations at once. This could make an actual sync operation simpler, but it's just a one-off in that it's specific to your situation and wouldn't likely work for many, or any, others. A real sync option would have to be more general.
I'm curious, though. How would you like to see a sync work?
For example, since your laptops are in different physical locations, how would they communicate? Are we talking push a button and the software talks to the central site, perhaps over HTTP? (That would require a server.) Are the laptops on the same network and thus could share drivers? Or would importing/exporting (more like backup/restore) the database and sending those files via email work?