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 the 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 the 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 the formation of four distinct families of vortex solitons with topological charges m = ±1, all of which can be stable. At the same time, only two stable families of fundamental gap solitons characterized by the domination of different polarization components are encountered.