There was a huge announcement today by the Exchange Online team for expanded support of delegation inside of OWA - http://blogs.office.com/b/office365tech/archive/2013/09/04/configuring-delegate-access-in-outlook-web-app.aspx.
I am asked this a lot as organizations have user bases that do not require the full Outlook experience to do email; they only need OWA. Plus OWA has been closing the gaps with Outlook. Traditionally if you wanted to do granular delegation, a user is required to use Outlook to do this delegation.
When the new Exchange Online was released in the cloud (Exchange 2013), it included the ability to delegate your calendar through the browser. This has been expanded to support delegation of email folders, like your Inbox. Plus this delegation feature is being supported in the new Outlook Web Apps for iPhone and iPad; which is great for people who need to work with Exchange Online across devices and platforms. Please read the post for details - http://blogs.office.com/b/office365tech/archive/2013/09/04/configuring-delegate-access-in-outlook-web-app.aspx.
Now if you still need to delegate more than just email and calendar, Exchange Online admins can delegate an entire user’s mailbox to another user. They can then use the “Open Another Mailbox…” feature in OWA to completely manage and do “Send As”. Please review my blog posting in July 2013 on this topic - http://www.astaticstate.com/2013/07/delegating-mailbox-through-browser.html.