Hi,
Do you means that ERP software send message on behalf of a user, however this message do not exist on user's sent folder?
Please ensure this message deliver great by check recipient's inbox.
If so, I find an article about "Messages that are sent by using the "Send As" and "Send on behalf" permissions are copied only to the Sent Items folder of the sender in an Exchange Server 2010 environment":
https://support.microsoft.com/en-us/kb/2632409
In this scenario, the message is only copied to the Sent Items folder in MB1. You cannot configure Exchange Server 2010 so that the message is copied to the Sent Items folder of both MB1 and MB2.
To configure the sent items behavior for shared mailboxes we use the Set-MailboxSentItemsConfiguration cmdlet. This is available in Exchange Server 2010 SP2 cu4 but not yet available in Exchange Server