Another thing to remember when waiting to recieve items in the Moogle Delivery Service after making a purchase on the Mogstation or adding bonus item codes from merchandise to your account, is that the item will not be delivered in game until you logout completely from the game. Not just back to the title screen, but completely from the game. So if you're logged into the game still when you make a purchase, the item will not be sent until you logout completely.

If it still hasn't arrived after logging out and logging back in, and your inbox is clear, there is an option in the Inbox to resend system item emails, as sometimes things get backlogged, and usually that will help (it can still take a few minutes to work though). So just have patience.