Do you have access to the database on the LAN from your program? If so, you could add your client table as a linked table into the server database and then do a simple INSERT INTO FROM (SELECT ....) statement that will copy the necessary tables from the linked table (the client) to the destination table (the server).
If this is not feasible, please provide more information on the setup.
From the information you have supplied, I cannot tell for sure what you are trying to do. Where is the Access database located, on your local machine or on the server?
If you are trying to insert data into a local Access database, you can do this directly in Access on your local machine.
If you are trying to insert data into an Access database on a server, then you should be doing this from an application which has secure access to the database on the server. Although this is possible by using linked tables from a local Access database to an Access database located on a server, it is not a good idea because of security issues.
A better alternative is to connect to the server based database using a local application with a connection string that has access to the database on the server.