Hi Jason, you can author Tableau content on the web with Tableau Online or Tableau Server.
But that is not integration. I want to integrate the Tableau complete IDE in website
Base on my understanding to your question, the thing you called 'IDE' or the Tableau's Desktop application
for worksheet/VIZ editor is already integrated in Tableau Server (a web base application).
On that you can create and make vizuals for Tableau.
Or if you mean you have your private website and have Tableau Editor itself, then its not possible to embed it.