My company's application has been using Xcentrisity BIS (the RM/COBOL variant) for over a decade with great success. Not only do we use it in a 'direct to browser' mode for the application's user interface, we also use BIS in a variety of web services situations, both SOAP and REST. We are presently in the midst of an automated user interface conversion project which will get us further along the road to using REST architecture.
REST is not a specification, but rather an architectural style. In it simplest form, URIs are used to identify resources, and the HTTP verbs (PUT, GET, UPDATE, DELETE) are used to implement CRUD (Create, Read, Update, Delete). It is not unusual to see URL-encoded payloads, but XML payloads are also used and are becoming more common. If all this sounds familiar, it is because the World Wide Web uses the REST architecture.
Perhaps you would like to describe a bit more what your goal is, so that the experts here can provide a more specific recommendation.
Tom Morrison
Hill Country Software