The only way to listen to WebSocket communication is to inject a WebSocket wrapper into the site's code and have it relay the messages to you. You're code should look something like this:
manifest.json
{
  ...
  "content_scripts": [
    {
      "matches": [
        "<website-url>",    
      ],
      "js": ["content/syringe.js"]
    }
  ],
  "web_accessible_resources": ["lib/socket-sniffer.js"]
}
content/syringe.js
var s = document.createElement('script');
s.src = browser.runtime.getURL('lib/socket-sniffer.js');
s.onload = function() {
    this.remove();
};
lib/socket-sniffer.js
(function() {
  var OrigWebSocket = window.WebSocket;
  var callWebSocket = OrigWebSocket.apply.bind(OrigWebSocket);
  var wsAddListener = OrigWebSocket.prototype.addEventListener;
  wsAddListener = wsAddListener.call.bind(wsAddListener);
  window.WebSocket = function WebSocket(url, protocols) {
    var ws;
    if (!(this instanceof WebSocket)) {
      // Called without 'new' (browsers will throw an error).
      ws = callWebSocket(this, arguments);
    } else if (arguments.length === 1) {
      ws = new OrigWebSocket(url);
    } else if (arguments.length >= 2) {
      ws = new OrigWebSocket(url, protocols);
    } else { // No arguments (browsers will throw an error)
      ws = new OrigWebSocket();
    }
    wsAddListener(ws, 'message', function(event) {
      console.log("Received:", event);
    });
    return ws;
  }.bind();
  window.WebSocket.prototype = OrigWebSocket.prototype;
  window.WebSocket.prototype.constructor = window.WebSocket;
  var wsSend = OrigWebSocket.prototype.send;
  wsSend = wsSend.apply.bind(wsSend);
  OrigWebSocket.prototype.send = function(data) {
    console.log("Sent:", data);
    return wsSend(this, arguments);
  };
})();
(document.head || document.documentElement).appendChild(s);
All credit for the above code obviously goes to the posters linked above. I have only copied tt here for convenience' sake.
Update:
Yes, it does matter that the WebSocket is within an iframe! By default, extensions are only loaded in the top-most frame. For it to load into the iframe, you must add "all_frames": true to your manifest.json:
manifest.json
{
  ...
  "content_scripts": [
    {
      "matches": [
        "<website-url>",    
      ],
      "all_frames": true,      
      "js": ["content/syringe.js"]
    }
  ],
  "web_accessible_resources": ["lib/socket-sniffer.js"]
}
If you'd like to read more, here's the documentation for all_frames.