I might be ignorant if there exists such functionality. Can I search for a friend on Stack Exchange with their email address?.
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If they're your friend they can tell you their username in person, therefore your question seems to imply they're not your friend. There's no reason at all for you to know who is your friend on Stack Exchange apart from to engage in voting fraud; so, I don't see the need for this feature. Each site should be about the quality of the questions and answers not who wrote them.
Stack Exchange is not a social network.
Actually, you can.
By taking the md5 hash of your friend's email address, you can search the data explorer for other users using the email they used for their gravatar. This isn't foolproof, of course, since the email they used for the gravatar can be different from their usual email.
Though see Ben's answer for why this isn't a particularly good idea. I'm not entirely sure why you can't just, you know, ask your friend for his Stack Overflow username.