AWS Api Gateway local testing/development
To use the AWS API Gateway locally, spin up SAM locally using the SAM CLI. There are however limits to what you can do with SAM, so you may not be able to do what you need to do locally. For example, websockets are still not supported (as of Sept 2020) as far as I know.
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SAM stands for Serverless Application Model. See the Quickstart Guide for Developing Serverless Apps using SAM.
Other links:
Running API Gateway Locally
AWS CLI (required to run the SAM CLI)
Install the SAM CLI
SAM CLI Reference
Unfortunately, API Gateway doesn't offer local testing/development solutions at the moment.