There's a pretty big gap between a browser-level source editor + compilation, and what I would call an IDE.
The browser IDE that I've heard most about would be Cloud9. Don't know if they have any API.
Cloud9 uses (and maintains) the Ace editor, which you could adapt to your needs. It's not an api though, just an editor.
From the list, the only thing I've used is Hackerrank. Their editor/compiler thing is okay, though I still find myself developing in IntelliJ/PyCharm and copying the result. If their editor has some kind of API for this, that's news to me, I'm only aware of competition API.