Yes, but only if your store is on Shopify Plus. Unfortunately, Shopify limits access to the gift card API to Shopify Plus stores only.

If your store is on Shopify Plus, you can select the "Automatically-Created Gift Card" Advocate reward type to send your Advocates a gift card instead of a discount code. One benefit to this is that Advocates can use more than one gift card at a time during checkout, while discount codes are limited to one per order.

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