The data controller popup is what shows up in the top right when someone loads an item that is a data controller, for instance Uber (Q101).

Current implementation

MediaWiki:Common.js, which is run for everyone accessing a page, loads MediaWiki:Access.js
`Access.js` checks whether the page is an instance of (P3) data controller (Q96)
if so, the Template:MailtoAccess template is run with parameter the qID
a `href` is created

Wishlist

Make the whole popup be a wikitext template, using the MediaWiki Template framework
A warning to check the email address that things are sent to (this would be a sneaky way to be a bad guy)
A button to copy to clipboard the mailto link (browsers make it difficult to copy this type of link)
An invitation to Tweet "I just asked my data from company ____, join me at ____"

Ask for access to be able to modify the MediaWiki namespace
better yet, implement your new version as a user script:
- go to your own Special:Mypage/common.js
- add a line to load your own script (see how it's done and get inspired by any other user's scripts, for instance User:Podehaye/common.js)
- copy/paste/modify MediaWiki:Access.js into this new script, to do what you want in parallel to the current version (so you have two popups!)
- tell an admin to take your script and make it sidewide