How do I embed (is that even the right word?) a whole 3rd-party website into a web application such that the user can access the full functionality of that particular website within the latter?
You can use iframe to embed any site or(url) into yours. But if response header of that url contains x-frame-options: deny, then it will show blank. This is done to let owner of a website decide if and where they want their site to be embed.