Interesting scenario!
Maybe this can be a starting point? https://www.copernica.com/en/blog/post/how-to-create-email-buttons-with-just-html-and-css
I do not think you can avoid navigation away, because email clients will probably not allow javascript to be executed inside of an email for security reasons.
The best achievable scenario i can think of, is a deeplink to a page, and put the approve/reject buttons inside that page. Or put one approve and one reject button in the mail and accept that it will do a popup.
I hope this helps.