Cannot initialize the data source object of OLE DB provider "Microsoft.ACE.OLEDB.12.0" for linked server
When you set up a linked server, you have the choice to re-use the credentials of the user, use a specific account or be anonymous.  When you tested it from the same machine, and SQL Server was running as a local account, it worked because it was able to re-use the admin privileges of you while you were logged-in.  

To get it to work in a network scenario, things get tricky.  If you are using AD, you need to use AD (principalName) to configure the Kerberos trust between the SQL Server machine and the machine hosting the MSAccess file.  This usually takes some fiddling to get it right.  This TechNet article is long-winded, but contains a lot of config steps and suggestions: http://support.microsoft.com/kb/319723.

Also, you will need to set the server back to use the NETWORKING SERVICE account again so you can grant it permissions.  On the other machines, your SQL Server will be using a hidden account like [domain]\[MachineName]$.  If the MS Access DB is on the same machine as your SQL Server, you still need to grant (NTFS) read/write permissions on the MSAccess DB (file) to the accounts for the SQL Server process and to the network users.