Assuming you have SQL Server or MySql installed on the LAN, and your database has been added to the server, it should be just a case of providing your software with the correct connection string to access the Server, just as you had to when you were developing the app on your development machine.
Hopefully, you have written your app carefully, and your connection string is stored in a configuration or settings file rather than hardcoded into your application, and if so it should be simple enough to change the string.
What string should you use? I don't know - we don't have access to the network, much less the Server instance, so we can't be specific.
This may help though:
Simple SQL Connection String Creation[
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