We study two-dimensional fundamental and vortex solitons in polariton condensates with spin-orbit coupling and Zeeman splitting evolving in square arrays of microcavity pillars. Due to repulsive excitonic nonlinearity such states are encountered in finite gaps in the spectrum of the periodic array. Spin-orbit coupling between two polarization components stemming from TE-TM energy splitting of the cavity photons acting together with Zeeman splitting lifts the degeneracy between vortex solitons with opposite topological charges and makes their density profiles different for a fixed energy. This results in formation of four distinct families of vortex solitons with topological charges 1 m = , all of which can be stable. At the same time, only two stable families of fundamental gap solitons characterized by domination of different polarization components are encountered.Formation of stable nonlinear excitations in the external periodic potentials is a problem attracting considerable attention in diverse areas of physics, including photonics and matter wave optics. Periodic potential may qualitatively change spatial dispersion, so that excitation of bright lattice solitons with unconventional shapes becomes possible under conditions where bright solitons simply do not exist without the potential. The properties of lattice solitons are well-studied in nonlinear optical settings, where peri- One of the most distinctive features of polariton condensate is the possibility to realize in it sufficiently strong spin-orbit coupling originating in the cavity-induced TE-TM splitting of the polariton energy levels, see, e.g. [6,8,13] and references therein. Among other effects, such a coupling mediates formation of unidirectional polaritonic edge states in truncated honeycomb potentials [8,21]. Nevertheless, the impact of spin-orbit coupling on formation of gap polariton solitons in periodic potentials was not considered yet. At the same time, it is known that in Bose-Einstein condensates spinorbit coupling (of completely different physical origin [22]) may drastically affect stability and symmetries of available nonlinear excitations in periodic potentials [23,24].The aim of this Letter is twofold. First, we introduce vortex polariton solitons in periodic potentials. Second, we study the impact of spin-orbit interactions, specific for polariton condensates, on all gap soliton states, including fundamental and vortex ones. In order to do this we use continuous conservative model describing evolution of exciton-polariton condensates in a square array of microcavity pillars and accounting for spin-dependent interactions and Zeeman splitting in the external magnetic field. We found that in this system the properties of vortex solitons depend on the sign of their topological charge and that four distinct stable vortex soliton families can co-exist for the same energy.We describe the evolution of polariton condensate in periodic potential by a system of dimensionless Gross-Pitaevskii equations for the spin-positive and spin-negativ...