Turning off customer notifications for a store:
In the app, click on the Account Settings gear, and then the Selling Channels link.
Click Edit on the store for which you wish to turn off notifications, then click on the Notifications tab.
Now, for the Shipment Notification Email Template drop-down, make sure it's set to "(don't send email)".
Make sure to Save your changes.
Turning off marketplace notifications for a specific order:
In the app, click on the Orders tab. Now, find the order for which you want to turn off the marketplace notification, and double-click on it or click its order number.
Scroll down to the Other Shipping Options area. Here, check the box next to "Do not notify marketplace when shipped."
This will prevent the marketplace from being notified, but if you have an e-mail template setup for the store, your customer will still be notified via an e-mail.
Don't forget to Save your changes.