It is not a browser issue. Settings->Widgets can work for one account but not another in the same browser. My ‘proper’ account cannot access widgets settings in any browser or operating system, but a throwaway account I created can. I couldn’t even login to this site with my ‘proper’ account either, I had to use this throwaway to make this post.
The screen I get when clicking Settings->Widgets looks like this: http://i.imgur.com/0QefDcW.png
Widgets are still working, and new widgets can be created by accessing https://twitter.com/settings/widgets/new, but existing widgets cannot be modified.
I had this problem 9 months ago, it was fixed and it worked for a while, but now it’s happening again. Here’s the original thread (now closed) discussing the issue: Cannot access https://twitter.com/settings/widgets