How do I call an AWS_IAM authorized API Gateway endpoint from AWS Lambda who's execution role has permission to do so?

The credentials for lambda's execution role can be retrieved from environment variables - AWS_SESSION_TOKEN, AWS_ACCESS_KEY_ID, AWS_SECRET_ACCESS_KEY.

You can use the above credentials to sign the requests. The docs for signing are here http://docs.aws.amazon.com/general/latest/gr/sigv4-signed-request-examples.html. You can also look at third party libraries that do this.

Note that since you have session token you'll need to handle it as described here http://docs.aws.amazon.com/IAM/latest/UserGuide/id_credentials_temp_use-resources.html#RequestWithSTS