If the program relies on a database, just set up a central database server and have your program connect to it.
That's basically it.
Several copies of your program on various computers can connect to the database and share information.
You have to get your system or database admin to set up the database on the server and assign permissions. If you are going to allow multiple copies of your program to use the database concurrently then you will have to build in some sort of interlock to prevent multiple users from updating the same records at the same time. You will connect to the database using a connection string that incorporates the network name of the database server.
A LAN is a local area network. This is not as simple as enabling a feature or installing a program. Generally in a "system" there are various components. I would assume that this enrollment system would have an application (web or fat client based), a database server, and the network that this application rides on. Setting up your LAN has nothing to do with the app or the database server. There is no way that anyone is going to be able to provide you with instructions on how to setup a LAN in this thread. It sounds like you may need some help from someone that can spend some time to design and implement your network prior to getting this system into production. Your network design is going to depend on your requirements. For example, is this application only accessed from the intranet, or will you have internet clients as well. Does the application and database server(s) need to be behind a firewall, the list of questions goes on and on...