Mogstation item not in mail, it's been 5 days.

[Avidria]

The longest I've waited was a few minutes, or the duration of a dungeon because my queue popped before the mail got there. Maybe a dumb question but do you have like, a purchase confirmation? I think you can check past purchases in Mog Station
If not it might be just waiting for support to actually answer unfortunately
Or second dumb question, it's been a while since I dealt with this but is your mailbox full? I vaguely remember there being some 20 mail limit or something where it just won't send you more til you clean it out

[Eldevern]

...

Fair question, we are limited to 20 mails or 20 items.