Add "ASP.NET Web Form" to list of project types
Divide code section into ASPX and Code Behind.
Create Web App project on the fly and build it. With Default.aspx page having the code.
Display Default.aspx page in Output section or Popup
Web Forms is still used in many production applications, and support is not going away any time soon. That alone should make this a high priority suggestion.
Web Forms is old hat, it would be wasted time adding support for it.
Shekhar Shete commented
Please add inbuilt jquery with older to the newest versions and also functionality to publish the web application/project