From the subreddit’s flair policy:

Who Gets Flair?

Only mods can assign User Flair, and User Flair is only for conservatives. Once you have a solid history of comments in /r/Conservative, and have been commenting in the subreddit for at least two weeks, that is the right time to request flair via the link at the bottom of this page.

If you are not even somewhat conservative, don't bother asking. We're going to check your comment history and we may ask you probing questions about your views. If you're not measurably conservative, don't ask for flair. We will defend our turf and you can always try again later when you've spent more time posting the subreddit, proving that you deserve flair.

If you request a flair that doesn't indicate that you're primarily conservative, don't be surprised if we reject it. We constantly get requests like Libertarian, Moderate, Centrist, and even Progressive Conservative: no, no, no, and no. AnCaps aren't conservatives: you're living a fairy tale when you think a society with no authority at all can function. If you ask for a terrible flair that isn't conservative, you're digging yourself into a hole and even if you ask for something different, you'll probably have to prove yourself to us.

Only conservatives get flairs, based on their comment history.