Most of you guys must have noticed that you do not have the option to configure Send on Behalf permission for shared mailboxes in O365. This can be done only using the PowerShell. The below command could be run on PowerShell to achieve this:
Set-Mailbox -Identity firstname.lastname@example.org -GrantSendOnBehalfTo testuser
where, ‘email@example.com’ is the shared mailbox and ‘testuser’ is the user account or mailbox for which the permission is assigned.
You can confirm the above operation by using the below Get-command
Get-Mailbox -Identity mailbox | FL GrantSendOnBehalfTo
Reference : https://technet.microsoft.com/en-us/library/jj919240(v=exchg.150).aspx