Yes, Klaviyo can integrate with a custom website hosted on Amazon. To do this, you will need to use Klaviyo's APIs to connect your website to Klaviyo.

I just haven’t spotted the API connection: https://connect.klaviyo.com/integrations/?search=amazon

This is my first time with this, so I might be having user tech issues lol

If you haven’t spotted the API connection yet, visit Klaviyo's API documentation at Klaviyo API Documentation, generate your API keys from Account > Settings > API Keys, and follow the guidelines to use endpoints like Subscribe Profiles to a List and Track Events, integrating these into your website's backend to make HTTP requests and adding the Klaviyo web tracking script from Integrations > Web to capture on-site events