If you are invoking the second Lambda from the first using Amazon API Gateway, then your Lambda will need to have access to the internet. Follow this guide to configure a NAT Gateway (last step).
regarding the VPC: in order to connect to your VPC and access resources there, the Lambdas must reside in the same region as your VPC and also be configured access to your VPC.
Please follow the steps provided in this AWS Guide: Configuring a Lambda Function to Access Resources in an Amazon VPC. This guide advises to use AWS CLI commands to do this and does not show how to configure it through the console.
You will need to be familiar with Amazon networking particulars (VPCs, Security Groups and Subnets), IAM security for the VPC and have a CLI environment setup. You are going to grant the Lambda Function access to this VPC using IDs and IAM execution roles via the CLI.