If you need user to manage the mailbox of an another user or sent out as them, you will need to grant the user permissions over that mailbox. Exchange Online allows the following permissions to be set up:
Read and Manage: Also know as Full Access, allows another user to view and manage the contents of the user’s mailbox. These permissions allows emails to be read, moved and deleted, but does not allow emails to be sent out as the user.
Send As: Allows one user to send as another. To the recipient, an email sent this way will appear to have been sent by the other user. The Sent Item will still be placed into the original user’s mailbox.
Send on behalf: Allows one user to send as another. To the recipient, an email sent this way will show both the original sending user and the user they were sending for. The Sent Item will still be placed into the original user’s mailbox.
NOTE: Permissions are added onto the source account. For example, if you want User A to send as or manage the mailbox of User B, then permissions will need to be added onto User B’s mailbox.
Apply Mailbox Permissions
Permissions can take a while to propagate. If you recently made these changes, please allow up to an hour for them to process.