The most common way to implement this UX pattern is to deliver the gift or the Gift Page link by email.
To make this work, you probably want an email newsletter provider to collect these emails, so that you can easily send them Newsletters later.
* Warning, some affiliate links ahead
Personally, I like MailerLite - It's low-cost and easy to use. The automations are quite robust. The paid plans also support site pop-ups for other notifications, which my clients use a lot. That feature was a butt-saver during lockdowns, masks, and weekly policy changes.
Other options include;
Many ( including me ) recommend leaning away from MailChimp, due to unfortunate changes in their pricing, as well as tech limitations, and business practices.
Try a few out, specifically look at;