Nope.
Once you receive an order, your customer receives the notification you have setup for them to receive from your shop -- Crowdship will not email them anything.
Similarly, once the order is fulfilled, a tracking number is added to your your order (and to the order on your platform, like Shopify, WooCommerce, or Magento, if you have one integrated) and your customer only receives the notifications you have setup for them to receive -- Crowdship will not send them any emails.
Additionally, Crowdship has an option to use an order fulfillment notification feature. If enabled, Crowdship will attempt to use your platform's notifications to notify your customer that their order has been fulfilled.
For example, on Shopify, if this option is enabled,
and when the Supplier(s) on Crowdship fulfill your order,
the tracking number will be automatically added to your order,
and now since the option is enabled, a notification will go out from your Shopify store notifying the customer of the tracking number.