There isn't anything special in the API that would let you do this. You could send the user directly to the ask a question page.
I fiddled with the FireFox pluginFireFox plugin I created to see if I could add the ability to post directly to the site to add an answer to a question or a comment if someone was already logged in. If I remember correctly there is some key that is required for the POST request and it wasn't obvious how that key was generated.