Is there a way to get the icon or the IconUrl of the site so I can put it in my app like the DisplayName?
The methods stated in the comments are great for pre-defined sites, but in the future there will be new sites that people may want to add to your application immediately (if you have the facility for user-defined sites).
What I did in my app is inspect the index page of the site and scour it for logos (like a favicon, or iPad icon). It works OK, but an official mechanism for this would be better (somewhere in the
/stats/ page perhaps)?
A policy about the use of said icon would also help, stating that you have to write it to your own hosting rather than hotlinking, etc.
(I'm posting this an an answer so that it will bump the question and hopefully get it some attention from the powers-that-be).