I finished the Trouble on the Homefront quest but the trophy hasn't unlocked. Is it a glitch? Do I have to finish the quest in a certain way to unlock the trophy? I don't have any DLCs installed.
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)