So you want to use Cloudflare SSL certificates with AWS API Gateway? Its easy!
Go to the Cloudflare dashboard and create an origin certificate.
Cloudflare -> SSL/TLS-> Origin Certificate -> Create Certificate
Open the AWS Certificate Manager and provide it with your certificate body and private key. The certificate manager will ask you for a certificate chain. You can get that certificate here.
Click Review and Import. You are now ready to use it in the AWS API Gateway.