I think that this should be do-able by somebody with expertise, and with time on their hands (i.e, not me!). Sage does it by simply piggy-backing on iPython (http://ipython.org/), which provides a full API for a web-based notebook.
One of the problems seems to me is that for whatever reason, Axiom is niche software, and doesn't have the user base of either Maxima or Sage. It might be hard to garner support for such projects as this.