Its a little more complicated ... but not too much..and has to be throughly planned
Do you have a server or a computer on the network where you can store the databases? If yes create a folder on the server and move all your databases in the folder, now give all your users read-write rights on the folder.
The first program in your app should set the path to the shared folder.
Convert your application to an executable and install on each users machine... Though the user could run your exe from the server, I would not recommend it, it would be slow.
Thats about it in a nutshell...