This is earned by putting boxes into the blue recyclers across the street from the store. There is a perk you can get that has employees recycle boxes. They will take the boxes across the street and recycle them for you. This will come over time. You can check your progress on the Manager Blackboard, under the Statistics section. Once you get more than 50 on "Total Boxes Recycled" you are good.
It's worth noting, if you have the baler, they boxes that are put into it will not count towards this achievement.
The achievement will unlock after you close your store for the night after you recycle your 50th box.