Check your mailbox isn't full, you can only receive 20 messages from Square Enix (the middle tab when viewing mailbox) so if that's already at 20, delete something, press "delivery request" at the top middle and give it a moment.

That being said, Mog Station times can vary wildly. Sometimes it's quick, sometimes it takes ages, especially for account wide items.

Just double check in your mogstation order history that you purchased for the right character and then assuming you have enough mailbox space, just wait, it'll arrive