Normally, when you kill the Boss and open the Chest, when it shows 'Mission Complete' , you should receive the Triumph "In the Shadows of the Kingdom of Sorrow".
This one is located under Triumphs->Vanguard->Raids->Crown of Sorrow and grants the Raid Emblem when you redeem the Triumph.
As for the Raid Jacket Code, you need to log into bungie.net , then search for the Bungie Rewards Page under your profile. You should find the Raid Jacket there as well as a button to send yourself an e-mail with the discount Code. Make sure you have a linked and verified e-mail.
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Thanks! I'm not used to this emblem being tied to Triumph thing. Much Appreciated!