You will already need to have delegate permissions to a shared Outlook mailbox to send on behalf of it.
1. To send on behalf of a shared mailbox, open a new message in the Outlook desktop app.
2. Click on the Options tab in the menu bar of the untitled message.
3. Click on From to add the From field to your email message.
4. Click the expansion arrow next to the new From field and select Other Email Address...
5. Type in the name of the shared mailbox and click OK. Your message will send from the shared mailbox and appear to the recipient with <<your name>> On Behalf Of <<mailbox name>>.
Once you have added the From field in the first message, it will appear in all new messages, you'll only need to select which mailbox you are sending from. The default is your own personal mailbox. If you reply to a message sent to a shared mailbox, it will automatically be from the shared mailbox.