How do I send gift information to my fulfillment network?
Developer documentation for 3PL / fulfillment network integration.
When an order is placed with Wrapped gift options, the customer information (gift option, gift message, sender, recipient, etc) will be stored on the order details page.
If you are working with a 3PL / external fulfillment partner who has access to pull order data from your store (through the Shopify API) - you can let them know where the information is stored, or refer them here, to this article.
Order attributes
Refer to the Shopify documentation for: Order.customAttributes.
Line-item properties
Refer to the Shopify documentation for: LineItem.customAttributes.