One more note: there was another thread about this, earlier, and I posted the same info. You have to select your Moogle and talk to him like you would any other NPC in order to trigger the quest. It doesn't trigger automatically upon entering your moghouse, or when you click the Mog House button on the menu.
Also, after several Vana days (several hours), you have to select your Moogle and talk to him again to end the quest and get your reward. Your moogle will not actually go anywhere while he's "on vacation." You can continue to use your Moghouse as normal, while you wait.
Hope this helps. :)