Hi Jonathan,
Sorry for the inconvenience caused by the issue. In order to narrow down the issue, I would like to collect the following information:
• Please verify if the message receiving from OWA has any problems. I understand that the user can’t send emails from OWA. Is that right?
• Are there any error message during the sending message? If so, could you please provide us with the screenshot for further troubleshooting?
• Is the problem recently appeared with the user?
• Are you using Office 365 after the service upgrade or pre-upgrade? Please check out Am I using Office 365 after the service upgrade?
• Have you checked to send/receive emails from OWA with Office 365 mailbox like user@domain.onmicrosoft.com without any issues?
• Which browser and operating system are you using? For more information: Supported browsers for Outlook Web App
• Have you run O365 desktop setup?
Moreover, you can refer to the following thread it seems might be similar recently server side issue: http://community.office365.com/en-us/forums/158/t/211166.aspx
Best regards,
Shyamal