CloudFormation template implementing Private network which can be used by Serverless to deploy Lambda into VPCs an maintaining internet access

lambda-vpc-internet-access-cloudformation.yml

# Add the following to your existing VPC CF stack

  # create 2 subnets, lambdas like to be in multiple subnets
  Private1:
    Type: AWS::EC2::Subnet
    Properties:
      VpcId: !Ref VPC
      AvailabilityZone: !Select [ 0, !GetAZs ]
      CidrBlock: !Ref Private1CIDR

  Private2:
    Type: AWS::EC2::Subnet
    Properties:
      VpcId: !Ref VPC
      AvailabilityZone: !Select [ 0, !GetAZs ]
      CidrBlock: !Ref Private2CIDR

  NATIP:
    Type: AWS::EC2::EIP
    Properties:
      Domain: vpc

  NatGateway:
    Type: AWS::EC2::NatGateway
    Properties:
      AllocationId: !GetAtt NATIP.AllocationId
      SubnetId: !Ref Subnet1  # PUBLIC SUBNET!

  PrivateRouteTable:
    Type: AWS::EC2::RouteTable
    Properties:
      VpcId: !Ref VPC
      Tags:
        - Key: Name
          Value: !Sub "${Name} Private (Lambda)"

  DefaultPrivateRoute:
    Type: AWS::EC2::Route
    Properties:
      RouteTableId: !Ref PrivateRouteTable
      DestinationCidrBlock: 0.0.0.0/0
      NatGatewayId: !Ref NatGateway

  Private1RouteTableAssociation:
    Type: AWS::EC2::SubnetRouteTableAssociation
    Properties:
      RouteTableId: !Ref PrivateRouteTable
      SubnetId: !Ref Private1

  Private2RouteTableAssociation:
    Type: AWS::EC2::SubnetRouteTableAssociation
    Properties:
      RouteTableId: !Ref PrivateRouteTable
      SubnetId: !Ref Private2

Outputs:
  PrivateSubnet1:
    Value: !Ref Private1
    Export:
      Name: !Sub "${Pipe}-PrivateSubnet1"
  PrivateSubnet2:
    Value: !Ref Private2
    Export:
      Name: !Sub "${Pipe}-PrivateSubnet2"

I think you're supposed to feed Subnet1 as an input variable, leaving that up to you.