Only certain characters come standard with a wrench. Your second partner in the first level, Jock, has this ability. When you see a smoking object, switch to a wrench carrying character with , and press to fix the object. Also, if you see a wrench on the ground, any character can pick it up with and fix smoking objects. To check to see if a character comes ready with a wrench, press to see what they're carrying.
If you're looking to get this achievement in a hurry, go to the Pursuing the Ark level in the Raiders of the Lost Ark and fix the engine that powers the lift in the center of the first section of that level. This engine is a bit different than the other ones throughout the game in that it works for about 10 seconds to power the lift and then breaks again, allowing you to fix it over and over.