Save email attachments from MS Outlook 2016 to SharePoint site

It was  announced last week at Ignite that the OneDrive for Business Sync Client will support SharePoint libraries: https://blogs.office.com/2016/09/26/sharepoint-online-sync-preview-headlines-ignite-announcements-for-onedrive/

To support you scenario, have the user setup the new sync client to sync the document library where they want to store the email attachments. Then they can save a local copy in the folder for the document library. The sync client will then take care of "uploading" the file to the SharePoint document libary.   

I'm not sure if you have access to Flow with your Office 365 subscription level, but there were several demos at Ignite showing how to use Flow to automaticilly save email attachments to SharePoint.  That could be another option.