Yes, it's possible. If I understand you correctly, you'll probably want to use this approach:
Run the apache server on a different port from the node.js server, and have the static server serve the chat page. Then, you have two main options for how to get data from the static page to the node.js server: either use XMLHttpRequest directly from the static page with CORS (you'll need it because you're running from different ports, but it's still possible to have CORS allow from different ports on the same domain but nothing else, so it can still be secure), or have an invisible iframe of the node.js page on the static page, and then send data to it with postMessage, and then the iframe (which is on the same port as the node.js server, as it's being served by node.js) will forward the data from the postMessage to the server with XMLHttpRequest
You can also do a proxy, but it won't be as good for this type of situation I think, because if Apache is running the proxy, it completely erases how node.js does well with comet and things, but if you run node.js as the main server proxying to Apache, it would be easier just to do everything with node, so I'm guessing you don't want that