That sounds like a great idea to enhance user social networking! So I googled it!
Looks like someone successfully done so here
back in 2015.
That sure looks easy, however all it is enabling html within posts, allowing users to copy&paste source code a tweet to a topic post box on your site, and then your (or the css they supplied) custom css will display it however you want.
I haven't tried this, and not so much familiar with twitter, but if a user can copy & paste that source code form each tweet, everything that link says seems to be viable and as simple as copy&paste url or img codes too!
Post this idea, in extension request forum!