Can anyone make a World of Warcraft addon that can override the natural music and use an internet site like www.Pandora.com which basically is internet radio where you can change to music you like.

Many people I bet would enjoy this type of addon to switch to their own music for situations.

Just turn off ingame music and run Winamp in the background. You can set Winamps keybindings which work 'through' WoW to change tracks, play, volume, etc.

And no, an addon can't do this in any case.  There is no way for an addon to connect to a web site.  Blizzard made it so there isn't to help prevent people from writing addon viruses and trojans.