EZ Fulfill does not send any emails to customers when an order is out for delivery or is delivered.  However, Shopify does have an integration with a few carriers, which you can enable in your Shopify admin.

Configure Shopify To Send Out For Delivery and Delivered Notifications

Go to the notifications section of your Shopify admin and scroll down to the Shipping section.  You'll see two notifications here: Out for delivery and Delivered.  Check the check boxes next to these options to enable them.

If you want to edit the email templates, click the notification names, which links to the email template editor.  Here is Shopify's documentation on editing email templates.

Supported Carriers

Shopify has default integrations with a few carriers based on location.  The configured Out for delivery and Delivered email notifications will work automatically with the default integrations that Shopify has. 

If your carrier is not integrated with Shopify directly, these email notifications will not work.  EZ Fulfill doesn't integrate directly with carriers at this time, and the delivery emails come directly from Shopify.

These integrations may change over time, so it's best to review Shopify's documentation here instead of listing which carriers they currently work with.