Is there a cloud-based reverse proxy solution in AWS?
You can use AWS API Gateway (documentation).
API Gateway helps developers deliver robust, secure and scalable mobile and web application backends. API Gateway allows developers to securely connect mobile and web applications to business logic hosted on AWS Lambda, APIs hosted on Amazon EC2, or other publicly addressable web services hosted inside or outside of AWS. With API Gateway, developers can create and operate APIs for their backend services without developing and maintaining infrastructure to handle authorization and access control, traffic management, monitoring and analytics, version management and software development kit (SDK) generation.
API Gateway now supports HTTP Proxy integration for pass-through resources, so you don't need to describe your payload and query params explicitly (which was required previously).
Take a look at Kong.
Besides being open-source, you can operate it using its internal RESTful API, and you can extend it with custom plugins.
Just to add to @EdwardSamuel's answer:
Yes, the there is, and the proper AWS service for this is Application Load Balancer.
However, I'm really curious if @EdwardSamuel's answer really works because I'm also interested in using API Gateway for this purpose. I'm aware of some limitations with API Gateway (content-length limit, etc.) but I'm not sure yet whether these limitations will make it unusable to, say, reverse proxy a WordPress blog.