Purchase a latch and padlock from your local hardware store. The ideal, universal latching system utilizes a simple U-bolt that passes through the eye of the latch and can be easily secured with a padlock.
Some padlocks come with keys; some require a combination to be entered on a numbered dial. Both work well.
The latch you select should be the largest one that you think will fit comfortably on the cooler.
Locate a spot where the door and cooler body meet that can accommodate the latch. One side of the latch must sit properly on the cooler body and connect properly with the other side, which must sit properly on the door's body.
Affix the latch to the cooler door and cooler body with a heavy-duty epoxy or glue, such as J-B Kwik, Krazy Glue or Gorilla Glue. Follow the instructions on the adhesive's package.
Drilling or using screws on a refrigerator is highly inadvisable, as puncturing its body in any way can cause it to malfunction.
Test the latch to make sure it is aligned properly and works well. Open the padlock and pass it through the eyehole on the latch.
Close the padlock. The cooler is now locked and secure.