The lattice construction of Euclidean path integrals has been a successful approach of deriving 2+1D field theory dualities with a U(1) gauge field. In this work, we generalize this lattice construction to dualities with non-Abelian gauge fields. We construct the Euclidean space-time lattice path integral for a theory with strongly interacting SO(3) vector bosons and Majorana fermions coupled to an SO(3) gauge field and derive an exact duality between this theory and the theory of a free Majorana fermion on the space-time lattice. We argue that this lattice duality implies the desired infrared duality between the field theory with an SO(3) vector critical boson coupled to an SO(3)1 Chern-Simons gauge theory, and a free massless Majorana fermion in 2+1D. We also generalize the lattice construction of dualities to models with O(3) gauge fields. © 2019 American Physical Society.