Yes, AMP has it's own built-in Newsletter/Email capture popup.  This email popup is designed to make sure that your AMP pages remain compliant with Google's AMP requirements, while also being able to collect emails and offer a coupon code to subscribers. We have an announcement here regarding our newsletter popup integration.  This AMP newsletter popup sends customer emails directly to your Shopify "Email Subscribers" customer list.   (Make sure you've enabled customer permissions here).

Where do emails subscribers from AMP pages end up?

The emails from AMP end up here, in your Shopify Email Subscribers Customer List.

Third-party integrations

Third-party email solutions like Klaviyo, Omnisend, and others also integrate with your Shopify "Email Subscribers" customer list, so that emails sent from AMP can also end up in your third-party app.  While this is not a "direct" integration, it is a fantastic solution for AMP.  Klaviyo has an announcement here regarding their integration.

What about directly adding a third-party popup on AMP pages?

Google doesn't allow custom javascript on AMP pages, and since third-party apps like Klaviyo use Javascript, there's no way to automatically import them onto the AMP pages, but we make it super easy to add your own newsletter popup on the AMP pages, and those emails also go to your Shopify "Email Subscribers" customer list.