Destroy the source of the Mutants

In order to finish this quest you need to go to the Military Base which is west of Vault 13. If you go here you'll need a really great lockpicking skill and the electronic lockpicks to bypass the electronic lock on the front door, get in, defeat a ton of mutants, and then finally face VanHaggen on the fourth level of the base. An easier way to get straight to VanHaggen is to arm yourself to the teeth and then find a mutant named Harry in Necropolis. If you enter the room he's standing in, he'll automatically start dialogue with you and you can ask him to take you to his superiors. You'll find yourself face to face with VanHaggen and will then have a chance to fight him. You'll still have to fight your way out of the base, but that's slightly easier than fighting your way in...

Once this quest is completed you have finished half of the second part of the main quest. You are given an invisible time limit of 10 game years to complete both halves. Once you have also completed Destroy the Mutant leader, you will be transported to Vault 13 where the ending cutscenes will play and you will have completed the game.