Yes, it's possible.
You'll need to replace the browser-server communication with direct API calls.
1. For calling JavaScript functions from the WebForms application use WebBrowser.Document.InvokeScript(). Note that you can only call global JavaScript functions this way so you will need to wrap the method calls in special global functions or use "eval".
http://stackoverflow.com/questions/7322420/calling-javascript-object-method-using-webbrowser-document-invokescript
You will need InvokeScript to send data (events) to the controls and to invoke specfic actions (switching a date, changing config, reloading data, etc.)
2. For calling C#/VB methods from within the HTML page use ObjectForScipting property and you will be able to access that object using window.external object from JavaScript.
http://stackoverflow.com/questions/3694028/invoke-c-sharp-code-from-javascript-in-a-document-in-a-webbrowser
You will need window.external object to handle the JavaScript events (drag and drop, etc.).
See also:
https://msdn.microsoft.com/en-us/library/a0746166.aspx