OSGi has a part of the specification called Remote Services. In a very short time, it works so that you can register services with special service properties and, based on the properties technologies, you must pick up your service and create an endpoint from them. This is not only about REST, but also about any technology that processes remote calls. You can find information in the OSGi Core specification in the Remote Services chapter.
Well, this is a specification, but who implements it? There are currently two large projects that I have tried. CXF DOSGi and Eclipse ECF. They offer several technologies that support the specification of remote services. CXF especially supports Jax-RS based on its implementation both on the server side and on the client side.
Since I did not want to use spring specific solutions inside OSGi, I did not use CXF at the end, but created my own solution. It is based on the Jersey and Remote Services specifications. When the OSGi Service is specified with service.exported.interfaces = * and service.exported.configs = org.everit.osgi.remote.jersey, it will create a rest endpoint along the path / rest / using the HttpService. Your kit does not have to be a wab, it can be a simple kit.
I should mention that if you provide your services through any of the remote service implementations, you should use Jax-RS annotations in the interface that your source class implements and expose your service based on that interface.
Instead of @Resource and @Component annotations inside OSGi, I suggest you use Blueprint (part of the OSGi specification), which is surprisingly similar to Spring. Apache Aries and Gemini Blueprint are currently implementing it. Using a drawing, you can easily create beans and connect them to each other. If you register the remote service this way, you can set any property using the drawing (like the bean property in spring applicationcontext.xml).
You can find an example application that I made at https://source.everit.biz/svn/everit-osgi/trunk/samples/jaxrs/ (user / passwd: guest / guest). There is a guide explaining how these samples can be run and developed at http://cookbook.everit.org
Hope the sample application helps you get started with the Remote Services Services specification section.
To learn how to use JPA and Injection (Blueprint), you should check the Compendium OSGi specification for features and find the implementation you need. I also made a sample project based on the project and hibernate-jpa, which you can find as a relative of the sample URL that I already provided.
Update
There is also a JAXRS extender implementation that I did at https://github.com/everit-org/osgi-remote-jersey . See the README documentation. It differs from the first in such a way that it works based on the properties of the whiteboard service.