You can earn trophies with used PS Vita games. While Sony's system is cool, it's been set up to make it difficult to earn trophies if you buy a pre-owned game, so difficult, in fact, when you play a used game, you get this message:
"You cannot earn trophies in this application. Only the first player to use an application can earn trophies. To earn trophies, you must link the Sony Entertainment Network account of the first player to use this application with the PS Vita system."
Website Pocket-Gamer has a step-by-step tutorial for you, so check it out if you want to earn trophies from used game.