How to grant network access to LocalSystem account?

Solution 1:

In a domain environment, you can grant access rights to computer accounts; this applies to processes running on those computers as LocalSystem or NetworkService (but not LocalService, which presents anonymous credentials on the network) when they connect to remote systems.

So, if you have a computer called MANGO, you'll have an Active Directory computer account called MANGO$, which you can grant permissions to.

Note: You can't do any of this in a workgroup environment; this applies only to domains.

Solution 2:

You don't. If you need a service to connect to remote files or other network services, then you want to have the service run as a named account, and on the remote machine, assign rights to that named account.

It would really be best if you full explain what you're trying to do - that way you'll get the best answers.