I believe that you are referring to a different issue than the one reported by the original poster. The root post refers to the "New Contact" button missing from Outlook online (OWA). The missing New Contact button would have no effect on existing contacts. That would be a different issue. Troubling, but different.
The reason that I make this clarification is to keep this post on track because I too am experiencing the missing New Contact button issue with Outlook.com via any web browser, but all existing contacts are still in place.
Two problems.
First, I had hundreds of contacts go missing.
During the process of trying to recover them I noticed ALL the buttons missing when I was trying to backup what was left. I have no New Contact, Edit or Manager buttons... No action buttons are present at all. So I cant backup, import, edit or new from the browser,.
This is the issue that brought me to the thread... I flagged up this happened at the same time my contacts disappeared just in case the two were linked.