Do you think Apple would approve a client app to such a chat network?
If web browsers weren't grandfathered in then they would definitely not be allowed on the App Store and the web probably wouldn't exist today.
Do they allow e-mail clients there?
They already do in the form of email clients, IRC clients, mastodon clients, etc, etc.
Just off the top of my head, Apple allows for apps supporting IRC, RSS and Mastodon protocols. The latter came about in 2016, when these topics were just as relevant as they are now. Adding another protocol doesn’t seem difficult.