Need to clarify if there was a metallic raceway used as a grounding conductor for the original feeder. If so, no separate grounding conductor was required (and none is required now as long as you reuse the continuous metallic raceway). If not, then the next question is whether there was a neutral-ground bond at the subpanel. There better be! Without it, that's an extremely dangerous system - any ground fault will energize all "grounded" metal on the system. That is completely unacceptable and has never been code compliant.
Regardless of how the panel was set up before, your new installation requires an equipment grounding conductor returning to the main panel. That can be a wire or a metallic raceway. There are no exceptions. AFCI devices are not rated or allowed to be used for ground fault protection in lieu of a grounding conductor, and GFCI devices can only be used this way on unmodified existing branch circuits. You must have a fault clearing path from the EGC to the system neutral. The installation is unsafe and not code compliant without it.