There are two methods for finding your Etsy order number.
Method 1: Etsy order confirmation email
When you place an order on Etsy, you should receive a confirmation email that contains the order number you can use to submit your design. See screenshot below.
Method 2: Etsy orders page
When you place an order on Etsy, and you setup an Etsy login, you can view a list of orders to find your order number. Navigate to the Etsy orders page and login here.
Find your order, and then click the "View Receipt" button on the right side of the order as seen in the screenshot below.
Your order number will be at the top left of the page.