I know it is possible to receive your Facebook messages via email.
I would like to reply by email, but Facebook doesn't allow me to do this and instead returns a delivery failure.
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Sign up to join this communityFacebook now has "Facebook email" (your vanity name after the www.facebook.com/ followed by @facebook.com). You can send to others at facebook using that address from your normal email client.
Update: Facebook removed said feature. See comments.
I'm not aware of any Facebook apps that provide the functionality you're after.
As far as I can tell, you can only reply to status updates via e-mail. That's what the link from JoshMock's answer refers to.
The best way to make this work would be an XMPP to email gateway. Then you could add facebook messages as if it were a normal POP or IMAP email account.
Bonus feature: auto-replace the reply-to address with an email address, for any matching name.
As of late last year you could if you set your options on Facebook to send you an email when you get a message. But they have a history of turning this feature on and off, from what I hear, so no guarantees.