If you look on the internet you find many example of API Gateway integrated with Lambda to insert items to a DynamoDB or put records in a SQS Queue. That is not always necessary, because API Gateway provides lots of different direct integrations with other AWS Services (called First Class Integrations). See my previous post about Lambdaless integrations

When integrating with a Lambda function you can have Proxy and HTTP Custom integration, with other AWS Services you can only choose the non-proxy type.

Proxy type forwards request and response as is, while the http custom requires that you configure mappings for request and response.