Of course we can do this, but is it alright to do so? Are there any downsides of this approach?
What do you mean by session data? What programming language on the server (if you are referring to session data on the server)?
If you are using PHP, then you'll need to just call a PHP file with AJAX and delete using unset() (such as unset($_SESSION['user_id']) what ever you need to. I see this as just as risky as doing it with a normal request.
There's no reason you can't do this. The only concern would be that you'd need to handle any page state changes (like removing the 'account' links, etc).