Anyone can access the jumpquest. There are quests that can get you to and through the jumpquest much more easily, though. The Map of Phantom Forest will let you easily get to the jump quest. The Marker of Heroism let's you use teleportation spots in the jump quest to easily get through it.
If you want to know which quests you HAVE to do, look up those items and figure out which quests they come from. But my recommendation is to do all of the quests, as the dialogue is actually very interesting.