Nope. You have to be able to download it from the marketplace. A code isn't going to magically pop up the content off the disc.
Even with a Silver you can still buy & DL stuff from the marketplace right? Even if you can't do it on the 360 per se I think you can purchase it through the marketplace here & then simply DL it when you hook up your 360. I've done that a couple of times,
though granted I've got a Gold membership.
As another poster wrote if you can wait for the GOTY edition that we all know is gonna come. Not only will it have all the DLC it's gonna be more stable.
Yes, Silver members can buy games and add-ons. The main thing that gold membership gets you is the ability to play online with other gamers, which is not an issue with the Fallout games since there is no online play.