To add to Chris' answer: There can't be a "native COBOL app server" in the general sense. COBOL is an ISO standard language with many implementations on different platforms, generally with vendor-specific extensions as well. No application server is going to host binaries generated from COBOL sources on all those platforms and implementations. What Chris describes are application servers for various flavors of Micro Focus COBOL.
Even then, none of those servers is "specifically for COBOL" in any useful sense. IIS (for .NET COBOL) and Tomcat, Glassfish, etc (for JVM COBOL) are obviously used with programs written in several languages. Even Enterprise Server is used with PL/I and on occasion other languages. The same can be said for non-Micro Focus application servers traditionally used to host COBOL applications, such as CICS.
Chris mentioned that you can create RESTful web services under IIS using COBOL and the .NET framework. Similar capabilities are available for JBM COBOL. Enterprise Server does not currently provide a RESTful web service feature.