Best Method to Switch Between Terms (Custom Taxonomy)

If I am reading your code and description correctly, you can simplify that to:

$taxonomy = 'delivery_option';
$do = get_the_terms($post->ID,$taxonomy);
if (!empty($do) && !is_wp_error($do)) {
  $do = array_shift($do); //  assuming a single value as per the description.
  $term = str_replace('-delivery','',$do->slug);
  get_template_part('includes/sidebar-part',$term);
} else {
  echo '<p>No custom include!</p>';
}

Note, I used WordPress’s get_template_part() instead of PHP’s include() as that is the more correct way to include template files.