We propose a local detection scheme for the Majorana zero mode (MZM) carried by a vison in Kitaev's chiral spin liquid (CSL) using scanning tunneling microscopy (STM). The STM introduces a single Majorana into the system through hole-charge injection and the Majorana interacts with the MZM to form a stable composite object. We derive the exact analytical expression of single-hole Green's function in the Mott insulating limit of Kitaev's model, and show that the differential conductance has split peaks, as a consequence of resonant tunneling through the vison-hole composite. The peak splitting turns out comparable to the Majorana gap in CSL, well within the reach of experimental observation.