Location: Laetny's Cleft
Petitioner: Jonasz, Sellsword
Unlocked: Complete Letting Off Steam
Reward: 80 Wyrrite, 20 Renown, 800 gil
In order to unlock this side quest you need to progress far enough into the main story to reach Out of the Shadow. The majority of Out of the Shadow consists of three sub-quests, all called Letting Off Steam. Complete the three parts of Letting Off Steam, unlock the next area, and visit it once to gain access to all of the side content, including Get Cart.
As you're walking along Laetny's Cleft you'll come to a pair of sellswords who were hired to fetch a cart away from some bandits. They aren't certain of their skills, however, and enlist Clive to do the work for them. The pair couldn't be sketchier if they tried, but... a side quest is a side quest.
Continue south down the trail and you'll quickly run into a gang of Desert Raiders, as well as a Desert Djinn. The Djinn is a heavy-weapon-wielding type, and you've probably run into his like a few times by now. No special moves, just hefty swings - and you can avoid them all with Titanic Block, if you don't like to dodge. Reinforcements will show up partway through the fight, including a magic user, so be ready for them.
You'll receive Steelsilk for beating the bandits. At the other end of the ruins you'll find some old boards that you can bust through to continue down the trail. Look to your left once you destroy the boards and you'll find the missing wagon. Inspect it to complete the quest. (Turns out Jonasz is rather rude.)