I picked this quest up. I guess if you open the door prior to picking up the quest, and killing the mobs inside, the quest is incompletable?
I can only surmise this is the case after looking at wowhead and scouring every inch inside the vault I’ve opened up.
The cores you need are static interactable objects, not drops. The monsters being alive shouldn’t matter. Have you definitely opened all 29 doors? There are three floors with 8 doors each above the ground floor. The cores seem to mostly show up on the highest floor.
It is possible leaving the instance and returning after picking up the quest might be necessary, similar to how the jeweler kit and anvil get spawned in for the annulet quest.