By default, assume no. The lack of any official integration guide should be a clear sign. Even saying that you reverse-engineer Codex for apps to pretend to be Codex makes it clear that this is not an officially endorsed thing to do

Codex is Open Source though, so I wonder at what stage me adding features to Codex is different from me starting a new project and using the subscription.

But I believe OpenAI does let you use their subscription in third parties, so not an issue anyway.