Tag Rev 2.0
This article explains how to set up and enable Auto-Replies (previously known as "Out of Office") using OWA.
What you will need:
1/ A licensed mailbox in Office 365
2/ Your Office 365 login credentials
3/ A computer with internet access and a web-browser
First, login into OWA using your own mailbox credentials (How to access OWA)
From the toolbar, Click the SETTINGS icon:

From the bottom of the SETTINGS window, click VIEW ALL OUTLOOK SETTINGS
Click AUTOMATIC REPLIES:

Turn on automatic replies here:

Set the time period, and compose your auto reply:

CLICK SAVE
Your Automatic Reply is now activated
TO TURN OFF AUTO-REPLIES
To turn off auto-replies, repeat the steps shown above. Then, at the Automatic Replies screen, click here to turn it off:

Other Topical Instructions:
How to access Outlook on the Web (Office 365)
How to open a Shared Mailbox in OWA (Office 365)
Reference Material