We study a system of microcavity pillars arranged into a kagome lattice. We show that polarization-dependent tunnel coupling of microcavity pillars leads to the emergence of the effective spin-orbit interaction consisting of the Dresselhaus and Rashba terms, similar to the case of polaritonic graphene studied earlier. The appearance of the effective spin-orbit interaction combined with the time-reversal symmetry breaking resulting from the application of the magnetic field leads to the nontrivial topological properties of the Bloch bundles of polaritonic wave function. These are manifested in the opening of the gap in the band structure and topological edge states localized on the boundary. Such states are analogs of the edge states arising in topological insulators. Our study of polarization properties of the edge states clearly demonstrates that opening of the gap is associated with the band inversion in the region of the Dirac points of the Brillouin zone where the two bands corresponding to polaritons of opposite polarizations meet. For one particular type of boundary we observe a highly nonlinear energy dispersion of the edge state which makes a polaritonic kagome lattice a promising system for observation of edge state solitons.