Can you explain your use-case? API Gateway itself can expose your service to the public. API Gateway supports Regional or Edge Optimized endpoints that are external facing. So why do you want to add ALB in front of API Gateway again. What are you planning to achieve with that architecture?
Can you try using the private IP from Network Interfaces attached to the API Gateway as the ALB Target. But the Private IP can change so you need to have some custom solution(may be lambda) to monitor the IP change.
you need to use HTTP integration https://docs.aws.amazon.com/apigateway/latest/developerguide/getting-started-http-integrations.html see the following guide https://aws.amazon.com/premiumsupport/knowledge-center/api-gateway-application-load-balancers/
Failed to expose AWS ECS Fargate Service via ALBasked 7 months ago
How to get traffic from a public API Gateway to a private one?Accepted Answerasked 5 months ago
Is it possible to use ALB/NLB infront of an API Gateway.asked 3 months ago
AWS Api Gateway: how to expose new services automatically?asked 9 months ago
GRPC API through AWS ALBAccepted Answerasked 5 months ago
Load Balancer and API GatewayAccepted Answerasked 10 months ago
hot to make ALB forward traffic to API Gatewayasked 20 days ago
Public ALB - NAT Gatewayasked 3 years ago
Forward traffic from NLB to ALB in httpsasked a month ago
Lambda+ALB vs Lambda+API GWAccepted Answerasked a year ago