In order to have a certain receive connector to be able to relay (other than local delivery: actually route the mail to the outside) perform the following powershell command:
Set-ReceiveConnector "SERVER\Receive Connector" -PermissionGroups AnonymousUsers
Above command enables “Anyonymous user” delivery.
Get-ReceiveConnector "SERVER\Receive Connector" | Add-ADPermission -User 'NT AUTHORITY\Anonymous Logon' -ExtendedRights MS-Exch-SMTP-Accept-Any-Recipient
Above command enables this receive connector to be able to relay to the next configured step in your Exchange server (probably out).
Hope this helps you!