I doubt it. ASP.NET runs on the server, and has no access to the client
requesting the data. If such a thing was possible, it would have to be done
using a client side technology, such as JavaScript.
Out of curiosity, why would you want to minimise the website to the tray
anyway?
LIke ICQ, Yahoo also has something similar, where they emulate a cut-down
version of Yahoo Messenger in a web browser window. As far as I know,
neither of these companies web-based instant messaging products can minimise
to the system tray though.
You need to build a Windows Form based client application to do what you are
interested in. A Windows Form application can make and receive HTTP
Requests and Responses using the .Net framework.