Vorticity in polariton systems has been traditionally investigated within the semiclassical mean-field approach. In this work, we show that the fully quantized picture describes richer dynamics for the vortex core in the limit of single particle excitation. The quantum correlations intrinsic to our formalism account for the emergence of a family of trajectories that differ from the circular paths known in macroscopic vorticity phenomena. An analysis of the system dynamics reveals a correspondence between these nontrivial trajectories and the light-matter correlations, which define the hybrid quasiparticles. Our results provide a direct criterium to benchmark light-matter entanglement in polaritonic systems. Finally, we propose a concrete experimental realization that can be achieved within the current capabilities of experimental condensed matter physics.