Hi LizaC,
Based on your description, I know that you cannot receive the emails with you customer domain.
I have checked the customer domain with you, but I found that there are two MX records in your domain register, but there is no MX records used for Office 365. It is may be the reason for the issue.
If you are using the Small Business plan (P plan), There are two options to you:
1. You can change you name server to Office 365, Please refer to the following link: http://onlinehelp.microsoft.com/en-us/office365-smallbusinesses/gg549203.aspx
2. If you don’t want to change the name server to Office 365, you need to add the DNS records to Office 365. For send/receive emails, you need to add the MX record to your domain register. I also suggest you add the TXT record for Exchange Online.
About how to get the DNS records information, please refer to the following steps:
Sign in Microsoft Online Portal (MOP)>click Domains under Management in Admin page>select the customer domain, click View DNS settings, you will get the DNS records information.
If you are using Enterprise plan (E plan), you need to add the DNS records to Office 365.
About how to gather the DNS records information from Office 365 and how to add the DNS records, please refer to the following link:
http://onlinehelp.microsoft.com/en-us/office365-enterprises/jj655360.aspx#Outlook
Thanks,
Anna Shi