A Virtual Element Method (VEM) for shear deformable shallow shells, employing uncoupled first order approximation for the rotations and displacements is presented. The key novelty of the proposed approach is the elimination of shear and membrane locking within the element by projecting the shear and membrane strains onto a constant space, in a manner analogous to selective reduced integration techniques in Finite Element Methods (FEM), while retaining the flexibility to accommodate arbitrary polygonal element shapes. This formulation allows the generation of elements with only 5 degrees of freedom per node (3 displacement, 2 rotation). There have been VEM formulations for shear deformable plates previously proposed, such ones with Mixed Interpolation of Tensorial Components (MITC), higher order interpolation of displacements or selective stabilisation, and Kirchhoff shell formulations, however this formulation is an extension of the plate formulation presented previously. The proposed element is tested with both thick and thin plates and on various polygonal meshes, and is compared to the equivalent reduced integration quadrilateral finite element as well as with previously published analytical and numerical results. The element is tested in static bending, where the proposed element performs well against the FEM, on a much wider range of mesh topologies. The element is also tested on the free vibration and linear buckling problems in which the element performed well again, especially outperforming the FEM in prediction of the mode 1 vibrational frequency. Dynamic problems are solved with Newmark Beta time integration which shows consistent and accurate results. These findings demonstrate the feasibility of VEM for shear-deformable shallow shells in solving a broad class of engineering problems and provide a foundation for future developments, including full shell and geometrically nonlinear formulations.