Found the fix, for anyone still looking.
Go to the tower, on the left of the Postmaster is the quest kiosk. Access the kiosk and select the bottom right option on the menu. This contains the starting mission of the questline, select it and it will remove the quest inventory full error. You can now play the mission.
Credit to Reddit user Natandty for finding this fix:
https://www.reddit.com/r/DestinyTheGame/comments/kdaryj/quest_inventory_full_cant_do_first_mission_on/gfxw6hf?utm_medium=android_app&utm_source=share&context=3
[quote]Found the fix, for anyone still looking.
Go to the tower, on the left of the Postmaster is the quest kiosk. Access the kiosk and select the bottom right option on the menu. This contains the starting mission of the questline, select it and it will remove the quest inventory full error. You can now play the mission.
Credit to Reddit user Natandty for finding this fix:
https://www.reddit.com/r/DestinyTheGame/comments/kdaryj/quest_inventory_full_cant_do_first_mission_on/gfxw6hf?utm_medium=android_app&utm_source=share&context=3[/quote]
[Edit] This worked for me, on PC. AFAIK it works universally.