How can I make my desktop application into an OpenID relying party?
It seems to me that you should present a small web browser window for the authentication to be completed. The interaction would not be going through your code.
I would not suggest using web scrapers to do this. In doing so, you place your code in between the user's server and the user, which is a breach of the covenant that the user's password is not seen by the relying party.
OAuth is the right technology to use for a desktop application, but it doesn't use the existing password ecosystem that OpenID has, which was not a part of the OP's question.