Outlook Web Apps in Office 365 has an option that helps administrator to find out mailbox usage. To check the mailbox usage, login to Office 365 as an administrator then click on the Admin link.
Click on the General settings available under the Outlook section.
In the Mailboxes Admin screen, double click on the mailbox for which you want to check the usage. You can also select the Mailbox and click on the Details link.
In the Details screen, scroll down to the Mailbox Usage section and expand it see the mailbox space used by that user.