Hi Andy,
According to this situation, my suggestion is that add the IPs of Office 365 FOPE (Forefront Online Protection for Exchange) into their Firewall white list.
You can find the IPs of Office 365 FOPE in following link:
http://technet.microsoft.com/en-us/library/hh510075.aspx
For more information of Office 365 URLs and IP address ranges, you can refer to the article below:
http://onlinehelp.microsoft.com/Office365-enterprises/hh373144.aspx
Thanks,
Wilfred Ying