You're certainly on solid ground in claiming that inbox is common enough to be accepted, ensiform. I don't have to look outside this forum to demonstrate that - the word was used 6 times by forumites before your query.
Interestingly, Chihuahua already allows outbox, and classes it as a common word. I haven't checked if it has ever actually been possible in a puzzle. I suspect the word got into the source list with the meaning "be a better boxer than", but when I flagged it as common, I was no doubt thinking of email.
Inbox and outbox are among those terms borrowed from the pre-computer office environment, like desktop and folder. The whole terminology of opening and closing files began as a metaphor from manila-bound collections of papers.
But it seems that in this case another term, in-tray (usually hyphenated or as two separate words), was more often used for the physical repository of incoming paperwork. And in-tray is still often used with that meaning, or in a figurative sense, for the tasks confronting a politician or business executive. In-basket is also used. Whoever first used inbox and outbox for components of an electronic mail system may have been influenced by the idea of a mailbox.
Anyhow, there's no doubt that inbox and inboxes should be allowed, and should be common words.