SageMaker provides a full machine learning development environment on AWS. It works with the Amazon SageMaker Python SDK, which allows Jupyter Notebooks to interact with the functionality. This also provides the path to using the Amazon SageMaker Feature Store.
Is there any REST API available for SageMaker? Say one wanted to create their own custom UI, but still use SageMaker features, is this possible?
Can it be done using the Amazon API Gateway?
Amazon API Gateway currently does not provide first-class integration for SageMaker. But you can use these services via AWS SDK. If you wish, you can embed the AWS SDK calls into a service, host on AWS (e.g. running on EC2 or as lambda functions) and use API gateway to expose your REST API.
Actually, SageMaker is not fundamentally different from any other AWS service from this aspect.