The API only returns the website link (jarroddixon.com), not the Twitter and GitHub usernames. You can check the results here. The same holds for the Users table in SEDE, so that's not an option either.
I think the best way is just to 'scrape' the user's profile page (which is public, so you don't need to be logged in to view it), parse the HTML and look for <a> elements in the <div> with class user-links (curiously, there are two of them, one is the parent of the other).