We demonstrate the existence of semivortex (SV) solitons, with vorticities 0 and 1 in the two components, in a two-dimensional (2D) fermionic spinor system under the action of the Rashba-type spin-orbit coupling in the combination with the Zeeman splitting (ZS). In the "heavy-atom"approximation, which was previously elaborated for the bosonic system, the usual kinetic energy is neglected, which gives rise to a linear spectrum with a bandgap. The model includes the effective Pauli self -repulsion with power 7/3, as produced by the density -functional theory of Fermi superfluids. In the general case, the inter -component contact repulsion is included too. We construct a family of gap solitons of the SV type populating the spectral bandgap. A stability region is identified for the SV solitons, by means of systematic simulations, in the parameter plane of the cross -repulsion strength and chemical potential. The stability region agrees with the prediction of the antiVakhitov-Kolokolov criterion, which is a relevant necessary stability condition for systems with self -repulsive nonlinearities. We also test the stability of the SV solitons against a sudden change of the ZS strength, which initiates robust oscillations in the spin state of the soliton due to transfer of particles between the system's components.