Hey Azmouse & Born - I had the very same question forever! Finally found an online free app called Kwout.
Whenever you see a tweet you'd like to capture, click on the tweet's "Details" button to bring it up into it's own page.
Then click "kwout" in your bookmarks bar. The app captures the tweet's URL and opens it up in the application.
From there you're prompted to "cut out" the part of the tweet image you want to capture. Once you do that, kwout generates a screenshot of the image and gives you several ways to share it.
For the purposes of this MyBB-style forum, I've found the easiest thing to do is to right-click on the captured screenshot and then just do "copy image address" (I'm on a Mac) -- for Windows, I believe it would be to just right-click on the captured screen image and go to "Properties" to get the image URL.
Lastly, I just insert the URL of the captured screenshot as an "image" here.
Hope this helps!