During the mission Paternity Test, you will meet up with Officer Wilson for Lunch! When she asks you to guess who the mailbox bandit was choose the option "An animal?" and you'll get the achievement!
If your still having trouble, here is a video guide with all the chapter 3 related achievements and collectibles!
The Mailbox Bandit Achievement Starts at 1:03! Timestamps for the other achievements and collectibles are in the description!