How can I embed a Search Engine into my website

I'm working on my dissertation and I have to add a feature to a pre-existing web app called MonkeyPuzzle, a tool used for argumentation analysis.

This feature will allow users to view information from another website such as Reddit or Facebook and use the data from the webpage in MonkeyPuzzle. The webpage must be viewable from the MonkeyPuzzle Web app, i.e. embedded in an iFrame. I need to display the website as it would normally be displayed, this includes CSS, images, and JavaScript if necessary.

I've researched lots of ways to implement this feature, including using a python proxy to grab the header of the files requested and display them in an iFrame, and web scraping the entire webpage and recompiling them in the window. These two are the most realistic options, however, I can't seem to get it to work properly. The Proxy sometimes gets blocked by Reddit (probably because of CORS), and the web scraper is not user-specific, the files get saved on the server and not the user's Temp files.

I've noticed that website like HideMyAss and other Web Proxies are able to embed Google (or a similar search engine) into their website, displaying them exactly how they look normally. Does anyone know how they do this?

Thanks in advance, if anyone even knows the name of the method they use to access google, I would be incredibly grateful.

P.S. I know that the Reddit API is available, however, this feature needs to be able to work for all websites, that is why I have tried proxying and web scraping.