We're migrating a website and they'd like to keep the gift card system they used because they also use it with their physical location (GiveX is the gift card company).
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Well , AKAIK - yes you can integrate third party gift card system with Magento 2 by using Magento 2 rest api.
Magento have define very robust Rest Api , which is easy to use , here i am sharing link for giftcard rest api of magento 2 - http://devdocs.magento.com/swagger/#!/giftCardAccountGiftCardAccountManagementV1
So you can create custom module which fetch the data from third party gift card system and synced up with magento 2 and in reverse manner from magento 2 gift card data to that third party system.
let me know if i escaped something , hope it helps