Weapon SDUs are quest rewards in the main storyline. As the first one is early on, I would hazard a guess that you generally play multiplayer or split screen. I don't know for sure but I'd make a guess that the game issued the weapon sdu to the host of the game only. A possible solution would be to host or play single player and hopefully the quests will activate for that character to complete.
If the character has played through on single player, I'd have to say it's a bug and unfortunately I can't think of any solutions.