function mymodule_entity_insert(DrupalCoreEntityEntityInterface $entity) {

  if ($entity instanceof Drupalcommerce_storeEntityStoreInterface) {

      $sm_id = 1; // I can see Shipping Method ID is 1 from your table screenshot
      $shipping_method = Drupalcommerce_shippingEntityShippingMethod::load($sm_id);
      $store_ids = $shipping_method->getStoreIds(); // get existing IDs
      $new_store_id = $entity->id(); // get new id
      $store_ids[] = $new_store_id; // add new to existing
      $shipping_method->setStoreIds($store_ids); // set
      $shipping_method->save();

  }

}

But if your question is to add already existing stores to the shipping method, then run this code only once:

      $sm_id = 1; // I can see Shipping Method ID is 1 from your table screenshot
      $shipping_method = Drupalcommerce_shippingEntityShippingMethod::load($sm_id);
      $store_ids = [1, 2, 3, 4, 5, 6]; // Array of all the store IDs that belong to this shipping method.
      $shipping_method->setStoreIds($store_ids); // set
      $shipping_method->save();

The better API solution

Thanks to 4k4's answer I was able to commeup with a more efficient solution.

function mymodule_entity_insert(DrupalCoreEntityEntityInterface $entity) {

  if ($entity instanceof Drupalcommerce_storeEntityStoreInterface) {

      $sm_id = 1; // I can see Shipping Method ID is 1 from your table screenshot
      $shipping_method = Drupalcommerce_shippingEntityShippingMethod::load($sm_id);
      $new_store_id = $entity->id(); // get new id
      $shipping_method->{'stores'}[] = $new_store_id; // appends new store to existing list.
      $shipping_method->save();

  }

}

Manual Solution

Quick follow up, do I have to load all the stores at once? When I have 1000+ stores and more, that seems like it will be a little resource intensive. Is it possible to just add the new store?

At this point, you wouldn't want to use the entity API and do Drupalcommerce_shippingEntityShippingMethod::load as this would contain all the 1000+ store IDs, you will have to manually DB query & insert:

function mymodule_entity_insert(DrupalCoreEntityEntityInterface $entity) {

  if ($entity instanceof Drupalcommerce_storeEntityStoreInterface) {

    $sm_id = 1; // Shipping Method ID
    $connection = Drupal::database();

    $query = $connection->select('commerce_shipping_method__stores', 'csms');
    $query->range(0, 1)
      ->orderBy('delta', 'DESC')
      ->condition('csms.entity_id', $sm_id)
      ->addField('csms', 'delta');
    $last_delta = $query->execute()->fetchField();
    $new_delta = $last_delta + 1;
    $new_store_id = $entity->id(); // new store ID

    $insert = $connection->insert('commerce_shipping_method__stores')
      ->fields([
        'entity_id' => $sm_id,
        'stores_target_id' => $new_store_id,
        'revision_id' => $sm_id,
        'delta' => $new_delta,
        'bundle' => 'commerce_shipping_method',
        'langcode' => 'en',
      ])
      ->execute();

    $storage = Drupal::entityTypeManager()
      ->getStorage('commerce_shipping_method');
    $storage->resetCache([$sm_id]); // Flush Cache

  }

}