API Gateway integration with DAX

I don't think it's going to be possible as DAX is deployed inside a VPC so even if it would be possible to call DAX directly from API gateway, it would be possible only for Private REST API.

If you don't use DAX for anything else, you could still implement the caching at rest Api level instead of using DAX

There isn't a way to achieve this without using a Lambda in between, that too a VPC Lambda which could very well defeat the purpose of using a caching system to reduce data access latencies. Using DynamoDB directly may be a better option here and using APIGW caching if applicable.