Does an aircraft carrier have a VOR onboard?
Aircraft carriers use a TACAN, which stands for Tactical Aid to Navigation.  (The Navy is really good at making acronyms that don't follow normal conventions for abbreviation)

Functionally the TACAN is like a VOR with DME, providing bearing to and from the ship, as well as range.  The system is not susceptible to rocking motion, and least not perceptibly.  (and if you are already familiar with VORs, a VORTAC is a landbased co-located VOR and TACAN station.)

And true, an enemy could pick up the signal.  For this reason carrier airwings need to prove themselves proficient in EMCON (Emission Controlled) launch and recovery operations before being certified ready for deployment.  During an EMCON recovery crews are told where to expect the ship, what the expected recovery course will be, and  will be given a marshalling lat/long.  

At the appointed time, and with no radio communication, formation flights will peel off the bottom of the holding stack, drop to break altitude,  head towards where the ship should be, and sequence themselves in with traffic in the pattern.  It is a pretty amazing thing when it works.