I am new to the world of GIS web services, but am wondering if there is a way to determine the web service endpoints that an existing web application embedded on a web page (silverlight) is using?
You can use Firebug for Firefox, Chrome Developer Tools, or Fiddler.
Firebug will be the easiest if you already have Firefox. Download and install Firebug, restart your browser, then open the Silverlight application. Press F12 to open firebug, go to the Net tab and enable it, then refresh the page. When the map is completely loaded, in the firebug window look for a line that starts with "GET MapServer?f=json". Once you've located that line, you can hover over it to get the full URL, and right click + copy URL to copy it. Instructions are almost the same with Chrome Developer tools.