The Free Tier for AWS provides developers with no-cost access to nearly everything AWS offers. Granted, it's subject to usage caps, so consider this route best for getting a taste of how things work. Still, even with those caps, you can accomplish a lot.
Amazon API Gateway is one API-related service offered with a free tier quota. The full, for-pay version of Amazon API Gateway allows you to build front-end APIs for applications built on EC2, AWS Lambda, or “any Web application,” with meta-tools like traffic management, API version control, and monitoring all part of the package.
The Free Tier allows you to receive up to 1 million API calls per month, so it's good for seeing whether the Amazon way fits with the APIs you want to create.