Publish Post After Click On A Link

When the post is created and set to pending, build a unique identifier for auto-publishing, for example:

$unique = md5( $post->post_content );
add_post_meta( $post->ID, '_auto_publish', $unique );

Now create a link for the email:

$link = get_permalink( $post->ID );
$link = add_query_arg(
    array(
        'autopublish' => $unique,
        'pid'         => $post->ID
    ),
    $link
);

Send this link in your email to the submitter’s address:

print "<$link>";

Then watch for the matching $_GET parameters when the recipient clicks the link:

if ( isset ( $_GET[ 'autopublish' ] )
    and isset ( $_GET[ 'pid' ] )
    and is_numeric( $_GET[ 'pid' ] )
    and $post = get_post( $_GET[ 'pid' ] )
    and $_GET[ 'autopublish' ] === get_post_meta( $post->ID, '_auto_publish', TRUE )
    )
{
    $post->post_status="publish";
    wp_update_post( $post );
    delete_post_meta( $post->ID, '_auto_publish' );
}