We'd be very excited to help you out with contributing to any areas of LiveOak!
LiveOak doesn't implement the JAX-RS specification, or any other REST frameworks.
The guts of the request handling is within the "container" module. liveoak/modules/container/src/main/java/io/liveoak/container/traversal at master · liveoak-io/liveoak · GitHub contains the code that's responsible for building up a series of "steps" where each step is one part of the URI path of the request. Then the TraversingResponder iterates through the generated steps, calling readMembers(), updateProperties(), createMember(), or deleteMember() as appropriate on the Resource.
No need to apologize, we have a lot of work to do on our documentation! Any tips or feedback on where we can specifically improve it would be greatly appreciated.