Proveit init() uses Object.assing() (MediaWiki:Gadget-ProveIt.js , line 103 [1] ) to store data which is not supported in Internet Explorer or in older browsers.[2]
[1] https://commons.wikimedia.org/w/index.php?title=MediaWiki:Gadget-ProveIt.js&oldid=249405179
[2] https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/assign
I found out that, on IE11, the ProveIt toolbar works by switching to "classic version" of ProveIt via user preferences in enWP. Click the box to checkmark that option and then press "save changes", and it works. Furthermore, if you checkmark both "ProveIt" and "ProveIt (classic version)", you will see both the "[P]" button and the classic, old version of ProveIt.
The latest ProveIt version crucially relies on the "atob" JavaScript method (https://developer.mozilla.org/en-US/docs/Web/API/WindowOrWorkerGlobalScope/atob) which is supported by all modern browsers, but Internet Explorer only supports it from version 10 onward. Adding support for older browsers now comes at an unacceptably high development cost. Unless strong reasons are given for adding support for old browsers, this task is unlikely to ever be resolved.
People stuck with an old browser can always fall back to "ProveIt classic" as pointed out by someone else above.