Calling on-prem Web API from AWS Lambda

I'm assuming that you want to keep your on premises API private rather than expose it to the Internet. If so:

You'll need to create a link between a VPC and your on premises network. The default VPC is fine but you probably want to create a new VPC with a specific IP address range that doesn't overlap with your on premises network.

You can link the two together using the managed VPN service or Direct Connect. It's quicker and easier to set up a VPN (less expensive too!); you can switch to Direct Connect later if you require more consistent bandwidth and latency.

Then, configure your Lambda function to run within the context of the VPC which will allow it to access resources in that VPC and on premises via the VPN (or Direct Connect) that you have.