CORS errors occur when a web page hosted on one domain tries to make a request to a server hosted on another domain. In this scenario, the registration form hosted on the static website is trying to make a request to the API Gateway API endpoint hosted on a different domain, which is causing the error. To resolve this error, the Access-Control-Allow-Origin header needs to be set to the domain from which the request is being made. In this case, the header is already set to www.example.com on the CloudFront distribution origin. Therefore, the solutions architect should enable the CORS setting on the API Gateway API endpoint and ensure that the API endpoint is configured to return all responses that have the Access-Control-Allow-Origin header set to www.example.com. This will allow the API endpoint to respond to requests from the static website without a CORS error.

https://aws.amazon.com/premiumsupport/knowledge-center/api-gateway-cors-errors/