Would I need a star topology with a dedicated ground plane?
One of the advantages of star grounding is that you define the current paths for some signals.  With a ground plane the current paths are up to the electrons and the voltage potential.  You may end up causing a voltage dip or surge for one component on the board at a location where a higher surge current passes near the ground pin during, for instance, a relay activation.

It may not matter for your board, you haven't provided enough information to determine that, but the answer to your general question is:

No, a ground plane does not replace a star grounding system in all cases, and the problems a star grounding system can solve are not all resolved with a ground plane.  In fact a ground plane may exacerbate certain types of problems that a star grounding system can solve.