Does bookvault send an email to my customer when a Shopify order is dispatched?

Yes! Bookvault can send emails to your customers through Shopify when an order is dispatched!
To do this, you need to enable this on your Shopify portal for the notification to be sent out.
If you're automatically fulfilling orders, you need to make sure the 'Notify customers of their shipment' is checked under Settings > Checkout > Order processing.
If you're manually fulfilling orders on a one-by-one basis, the checkbox to 'Notify Customers' needs to be selected when requesting fulfilment.
It's also worth noting, Bookvault has not control over the look of these emails, as it's simply providing Shopify with the information to send them. The appearance can be changed on the Shopify dashboard in the notification settings (Settings > Shipping confirmation)