Access to folder is denied (log shipping)
It would be useful to see the output of the actual job itself.

You mentioned you are using local system, if so then you need to grant permissions to the computer account on the folder.  The account you give permissions to would be in the format:

&lt;DomainName&gt;\&lt;ServerName&gt;$


So for a computer called ServerA on the Contoso domain you would give permissions to the folder for this account

Contoso\ServerA$


The $ on the end means this is a computer account.

If this works you may still run into issues with logging  if the computer accounts don't have any permissions in SQL Server.