Create System Email for Gift (not card)

Is there a way to set up a system email that is sent to the recipient of a gift? As I understand it, nothing is sent to the ship-to email and our fulfillment center for certain items is no longer doing inserts or paper. We’d like a system email with the gift message and a nice “you have a gift coming” notification to be triggered by the “this is a gift”. I see nothing in action email set up that will allow me to do this.