Fallout 3 Trouble on the Homefront trophy isn't unlocking

Based on the comments of the post you have encountered a glitch in the game. After leaving Vault 101 the quest should progress and the task to "Leave the Vault forever" should complete. There are no other requirements to earn the achievement apart from finishing the quest.

The only solution to this would be to replay the quest again using an old save or restarting the quest using console commands (which are unavailable on the PS3 and Xbox 360 unfortunately)