Instructions
Watch for wild Melynxes and Felynes when you go on quests. These are both cat-like monsters. If you've been to the Pokke Farm, you should have met a "domesticated" Felyne named Trenya.
Download the additional quest, "Two Colors of Meow" from the PlayStation Store. This quest has the highest concentration of Melynxes and Felynes, so your chances of finding Pawprint Stamps are increased. Although you will find these monsters in the wild, Pawprint Stamps are very rare drops.
Fight as many Melynxes and Felynes as you can. More often than not, these monsters will retreat by burrowing underground. Occasionally, you will see something shiny above the hole they leave behind. Inspect the shiny object and it may be a Pawprint Stamp.
Go to the Pokke Farm and speak with Trenya. Send him on an adventure to the Forest and Hills or Volcano region. Pay him either 500 or 1,000 Pokke Points and he will sometimes bring back Pawprint Stamps.