The Outlook connection is strictly an Outlook client feature. This creates a seperate PST file on the local machine and none of the information is stored in your mailbox.
Assuming you have access to IE in order to access OWA, in O365 you can then simply access the SharePoint site/collection to access the list.
The other option you have is to copy the contacts from the Outlook connection to SharePoint into your mailbox's contact list. There are 3rd party add-ins that will do this for you, but I'm not familiar with who produces them.
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