Yes, you can post extensions here that don't use the API.  There are several scripts/extensions/apps posted here that use [undocumented interfaces<sup>(example)</sup>][1] or even [screen scraping<sup>(example)</sup>][2].  

The extension [must make use of the API and/**or enhance one or more Stack Exchange sites**][3].

Reference:

 1. [Stack Exchange APIs Terms of Use][5] (Still applies, for the most part)
 4. [Etiquette of Screen-scraping Stack Overflow?][7]
 2. [Stack Exchange Network Terms of Service][4]
 3. [Guidelines for the Use of the Stack Exchange Trademarks][6]

----------

You should use the API for your data as much as possible.
---------------------------------------------------------
 
This *theoretically* avoids undocumented changes and staves off the ban-hammer.

In this case, you are fetching unread counts of notifications and inbox items.  
There are existing API routes for both:

 - [/inbox/unread][8]  
  and
 - [/notifications/unread][9] 

Ideally you would use these and not [that URL][10].  Although it can be quite a bother in this case, since both routes require OAuth and an `access_token`.  
However, the URL might also yield unexpected results if the extension user is not authenticated and depending on how the extension calls it.


  [1]: http://stackapps.com/questions/5056/hat-overflow-show-number-of-winterbash-hats-at-a-glance
  [2]: http://stackapps.com/q/2982/7653
  [3]: http://stackapps.com/help/on-topic
  [4]: http://stackexchange.com/legal/terms-of-service
  [5]: http://stackexchange.com/legal/api-terms-of-use
  [6]: http://stackexchange.com/legal/trademark-guidance
  [7]: http://meta.stackexchange.com/questions/443/etiquette-of-screen-scraping-stack-overflow
  [8]: https://api.stackexchange.com/docs/inbox-unread
  [9]: https://api.stackexchange.com/docs/unread-notifications
  [10]: http://stackexchange.com/topbar/get-unread-counts