Stealing the boxes can be done by waiting until 10PM, in game time and then all of the shops will close. Once they are closed, boxes appear in the safes of most shops. With the exception of Freckle Bitches, The Salon and Rim Jobs.
Simply break in to these shops and steal the boxes and then load them up into a vehicle. This will trigger the thief activity.
An alternate, and I believe superior method you can use to obtain this achievement would be to use the FBI vehicle. It can store up to six boxes and is relatively fast and agile. Once all six boxes have been obtained, drop them off and repeat the process at different stores.
To speed up the process, use taxi's to waste in-game time. You can kill 3 hours per trip so it doesn't take long. Using the FBI vehicle you can get about 12 boxes a night, just be fast and if you have too much heat, drop the boxes off early.