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I'd take a look here:
It sounds like you have a field mapping issue. Do you have access to Outlook? If so I'd recommend setting up your profile in outlook and importing there, it will work much better than doing it through OWA I think.
Jorge R. Diaz
Microsoft MVP | Office 365
MCITP: Office 365 Administrator
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Are you importing them directly to Outlook or into Exchange as external contacts?
Jorge's reply can greatly help us narrow down the issue. Please answer the above question in your next reply. Moreover, I would like to confirm some thing others with you:
1. Are you importing contacts to a common mailbox in Office 365, or a shared mailbox?
2. Is your Office 365 mailbox using a separate Outlook profile or share profile with other mailboxes?
If the Office 365 mailbox is not using a separate profile, I suggest you create a new profile for Office 365 mailbox, export the contact information to a new .CSV file to see if the format is different from the previous one.
Here are some general steps about exporting and importing contacts in Outlook 2010:
If the issue persists, please answer the above questions and I will continue to do more troubleshooting with you.
I am importing the .CSV file to the Outlook Web App on Office 365, and it is a separate, personal mailbox. When exporting contacts from my Desktop Outlook, the file looked the same as the CardScan .CSV file.
Have you tried to import the CSV file with desktop Outlook as Jorge suggested? Please let us know if you need further help.