Why is vault 75 quest not activating upon discovery?

Solution 1:

It is a known bug (mild spoiler alert):

"The Gunner that holds the Vault 75 admin access card may not spawn when you first explore the area. The location where he spawns is above the target practice area where you also find the bobblehead. Moving further away from the area or leaving Vault 75 and coming back will spawn the Gunner as normal. If the quest is active the marker will indicate his location."

Note that apart from in the inventory of the Gunner commander, the access card may spawn in the toolkit in the same room.

If you're playing on PC I suggest using the console command Player.AddItem 001a58a2 1, which will spawn the card into your inventory, so you can access the Vault 75 lab.

On consoles, you can try fast travelling elsewhere, waiting for (up to) 20 days, and returning (or keep playing the game without revisiting the area for that amount of time), or load an older saved game.