An AWS CloudFormation template for creating an API using API Gateway, with an EC2 backend. As an example, API Gateway's /api_endpoint points to the EC2's /ec2_endpoint.

apigateway_with_ec2.yaml

AWSTemplateFormatVersion: 2010-09-09
Transform: AWS::Serverless-2016-10-31

Parameters:

  VpcId:
    Type: String
    Description: The ID of the VPC containing the EC2 instance

  InstanceAZ:
    Type: String
    Description: The availability zone containing the EC2 instance

  InstanceId:
    Type: String
    Description: The ID of the EC2 instance

  InstancePort:
    Type: String
    Description: The port on which the EC2 instance is listening

  Subnet:
    Type: String
    Description: The ID of the subnet containing the EC2 instance
  
Resources:

  #############################
  # VPC
  #############################

  TargetGroup:
    Type: AWS::ElasticLoadBalancingV2::TargetGroup
    Properties: 
      Port: 80
      Protocol: TCP
      TargetType: instance
      Targets: 
        - Id: !Ref InstanceId
          Port: !Ref InstancePort
      VpcId: !Ref VpcId

  LoadBalancer:
    Type: AWS::ElasticLoadBalancingV2::LoadBalancer
    Properties:
      Scheme: internal
      Subnets: 
        - !Ref Subnet
      Type: network

  Listener:
    Type: AWS::ElasticLoadBalancingV2::Listener
    Properties: 
      DefaultActions: 
        - TargetGroupArn: !Ref TargetGroup
          Type: forward
      LoadBalancerArn: !Ref LoadBalancer
      Port: 80
      Protocol: TCP

  VpcLink:
    Type: AWS::ApiGateway::VpcLink
    Properties: 
      Name: vpc-link
      TargetArns: 
        - !Ref LoadBalancer

  #############################
  # API
  #############################

  Api:
    Type: AWS::Serverless::Api
    Properties:
      StageName: production
      Cors:
        AllowMethods: "'GET,POST,OPTIONS'"
        AllowHeaders: "'Content-Type,X-Amz-Date,Authorization,X-Api-Key,X-Amz-Security-Token'"
        AllowOrigin: "'*'"
      DefinitionBody:
        swagger: 2.0
        paths:
          /api_endpoint:
            get:
              x-amazon-apigateway-integration:
                connectionId: !Ref VpcLink
                connectionType: VPC_LINK
                httpMethod: GET
                type: http_proxy
                uri: !Join ["", ["http://", !GetAtt LoadBalancer.DNSName, "/ec2_endpoint"]]

deploy.sh

REGION= # Enter a region, e.g. us-east-1
STACK= # Enter a name for this stack
BUCKET= # Enter a name for an S3 bucket to use to deploy this stack

VPC_ID= # Enter ID of the VPC containing the EC2 instance
INSTANCE_AZ = # Enter Availability zone containing the EC2 instance
INSTANCE_ID = # Enter ID of the EC2 instance
INSTANCE_PORT= # Enter the port on which the EC2 instance is listening
SUBNET= # Enter the ID of the subnet containing the EC2 instance

aws s3 mb s3://$BUCKET --region $REGION

aws cloudformation package --template-file apigateway_with_ec2.yaml \
                           --output-template-file package.yaml \
                           --s3-bucket $BUCKET

aws cloudformation deploy --template-file package.yaml \
                          --stack-name $STACK \
                          --region $REGION \
                          --capabilities CAPABILITY_NAMED_IAM \
                          --parameter-overrides \
                            VpcId=$VPC_ID \
                            InstanceAZ=$INSTANCE_AZ \
                            InstanceId=$INSTANCE_ID \
                            InstancePort=$INSTANCE_PORT \
                            Subnet=$SUBNET

rm package.yaml