Ajax – Call to undefined function get_option()

You’re firing a request to a PHP file that won’t have the WordPress library loaded – hence the undefined function error.

You could manually load in WordPress with something like require '../../../wp-load.php'.

Or, better yet, use the awesome AJAX API. Just fire your request to wp-admin/admin-ajax.php and then register a callback in your plugin or theme that matches the action parameter (in your case, load_blog_posts_sidebar):

function wpse_147692_load_blog_posts_sidebar() {
    // Do something!
    exit;
}

add_action( 'wp_ajax_nopriv_load_blog_posts_sidebar', 'wpse_147692_load_blog_posts_sidebar' );
add_action( 'wp_ajax_load_blog_posts_sidebar',        'wpse_147692_load_blog_posts_sidebar' );