Stealing cookies from a web browser is a standard thing when you're writing a scraping script and don't want to deal with some complex website authentication in your script.
Traditionally, browsers stored their cookies in a simple text format inherited by Netscape, the
cookies.txt format. Wget has an option
--load-cookies to load a file like this.
Modern versions of Firefox and Chrome no longer use this format, but rather an SQLite database. There are extensions to make them dump a
cookies.txt file, such as Cookie Exporter and cookie.txt export.
This is all very well for a script for use by technical-minded people, but I wouldn't do this in a general-public application. You are in for a world of hurt figuring out what browser to use (this user has three Firefox profiles and two Chrome profiles, which one should I look at?) and making users of less common browsers angry (Opera, Konqueror, Gecko, …). It's also a big hit on usability, even when it works: you're asking the user to do something in their browser before they can use your application, and not to do something in their browser (logging out of the site or erasing cookies) as long as they keep using your application.
I would still consider the approach reasonable if the target website was very hard to interact with. And even then, I would favor making your application interact with its own browser instance (using its own profile) with a browser automation framework such as Selenium or Watir. For a general OpenID login, I think it's best to invoke a known browser and make it spit out the authentication credentials.