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I will lay out it for the uninformed
Twitter owns that website.
As owners of the website they have final control over what appears there.
Twitter can block users from tweeting and even close someone's Titter account.
The WOT ratings appear after every tweet.
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Likewise I will lay it out for the uninformed.
The WOT code doesn't appear on the Twitter web site but on your Browser, because you have installed the 'browser' add-on.
So the code is inserted into the page layout in your browsers display of the page. You can't change the sites source code on the fly at site level as that would breach the general site security protocols. Much of a sites content is only collated on the fly using content management software, so you wouldn't have access to that content anyway.
Don't believe me, then try this simple test.
Run firefox with WOT disabled and visit one of the twitter pages:
1. View the page source code of the page (which is actually in the browser cache) and see if there is anything relating to the WOT ratings.
2. Now run firefox with the WOT add-on enabled and check the page source code.
I somehow doubt WOT as every single domain in the world for permission to do this, check this old avast twitter page and there are WOT icons, I doubt they asked.