Execute certain function on GET request

One of the earlier hooks that is aware of the page (I think) is load-{$slug}:

add_action( 'load-_my-plugin', 'wpse143915_maybe_trigger_export' );
function wpse143915_maybe_trigger_export(){
    if( isset( $_GET['export'] ) && ( $_GET['export'] == 'data' ) ){
        //...
    }
}

If you want to find out what your page slug is, checkout this plug-in by Kaiser: https://github.com/franz-josef-kaiser/current-admin-info

But personally, I wouldn’t bother waiting for the (admin) screen to be initalised. Just send them to admin-post.php?action=myprefix-export, and then you know when that url is hit:

add_action( 'admin_post_myprefix-export', 'wpse143915_initiate_export' );
function wpse143915_initiate_export(){
   //Any security / validation checks

   //... export data...
}