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In this study, we calculate the nuclear matrix elements (NMEs) for the light neutrino-exchange mechanism of neutrinoless double beta $0\nu\beta\beta$) decay of $^{124}$Sn within the framework of the interacting nuclear shell model using the effective shell model Hamiltonian GCN5082. A novel method based on a nonclosure approach is employed, wherein for the intermediate nucleus $^{124}$Sb, effects of energy of 100 states for each $J_{k}^{\pi}$=$0^{+}$ to $11^{+}$ and $2^{-}$ to $9^{-}$ ($\Delta J_{k}$=1) are explicitly included in the NMEs calculation. Other common effects such as the finite size of nucleons, higher-order effects of nucleon currents, and short-range correlations (SRC) of nucleons are also taken into account. The extracted optimal closure energy is 2.9 MeV for a total NME of $^{124}$Sn $0\nu\beta\beta$ decay, which is independent of different forms of SRC parametrizations. A comparison of NMEs and half-lives with some of the recent calculations is presented. Further, to gain a comprehensive understanding of the role of nuclear structure on the $0\nu\beta\beta$ decay, the dependence of NMEs on spin-parity of the intermediate states, coupled spin-parity of neutrons and protons, and the number of intermediate states, is explored. It is observed that the inclusion of the effects of excitation energies of the intermediate nucleus yields more reliable NMEs. The present findings provide valuable insights for experimental investigations of $0\nu\beta\beta$ decay of $^{124}$Sn in India and elsewhere.