Kinetic Scout Rifle - 1 CRC
Kinetic Submachine Gun - 2 CRC & 1 APA
Energy Hand Cannon - 1 CRC & 2 APA
Kinetic Pulse Rifle - 3 CRC
Energy Sidearm - 3 APA & 1 FHB
Energy Submachine Gun - 2 CRC & 2 APA
Kinetic Hand Cannon - 1 APA & 2 FHB
Energy Pulse Rifle - 2 APA & 2 FHB
Energy Auto Rifle - 2 CRC & 2 APA & 2 FHB
Sword - 3 APA
Shotgun - Legends Lost Quest Line from Brother Vance
CRC - Concentrated Radiolarian Culture
Gather 10 Radiolarian Culture from public events and chests. Give to Brother Vance to upgrade.
APA - Advanced Paradox Amplifier
Gather 10 Paradox Amplifiers from completing strikes, Crucible matches, and Heroic adventures on Mercury. Give to Brother Vance to upgrade.
Heroic Strikes also reward Paradox Amplifiers. After testing this last night, there were strikes where I received both Paradox Amplifiers and Fossilized Hermaion Blossom so it is definitely more rewarding to stick to heroic strikes.
FHB - Fossilized Hermaion Blossom
Acquired by completing Nightfall strikes, Trials matches, and Heroic strikes.
How do you get these weapons?
As you go through the CoO campaign, Brother Vance will eventually send you to complete a Heroic Adventure. Once it's completed, he will give you Lost Prophecy, Verse 1. Grind out the materials, return to Vance, and forge the first weapon. After Verse 1, Vance will offer you a choice between Verses 3, 4, and 6. Once all three are forged, more Verses will be offered. I've completed only 1 and 3 so I will update as I go with future choice sets.