Lambda to ECS Integration in a VPC

Hello.

First, you need to be able to connect to the ECS endpoint from a Lambda connected to the VPC.
So please tie Lambda to the private subnet of the VPC.
Then add a route to the NAT Gateway to the private subnet's route table or add an ECS VPC endpoint.
https://docs.aws.amazon.com/lambda/latest/dg/foundation-networking.html
https://docs.aws.amazon.com/AmazonECS/latest/developerguide/vpc-endpoints.html

Also, be sure to properly configure the security group tied to Lambda and the ECS security group to access containers launched on the ECS from Lambda.