Drilling rotation is a key ingredient in Mindlin-Reissner shell formulations, as it enables a natural coupling between membrane and bending behaviours. The classical Allman-type kinematic assumption is commonly used to incorporate drilling rotation, but it fails to represent the actual rotational field and introduces a well‑known spurious mode. While several remedies have been proposed for triangular and quadrilateral finite elements, an effective and robust treatment of drilling rotation for polygonal discretizations is still lacking. This work introduces a polygonal shell virtual element with vertex drilling degrees of freedom that is free form spurious modes and does not require stabilization, thanks to a Hybrid Virtual Element framework. The element features six degrees of freedom per vertex and defines displacement and rotation fields only along the element boundary, while internal behaviour is represented through a-priori equilibrated stress resultants. This work presents a polygonal shell Virtual Element Method (VEM) featuring drilling degrees of freedom that is free from spurious modes and does not require any stabilization. The Hybrid Virtual Element (HVE) framework is adopted, defining displacement and rotation fields only along the element boundary, while the internal behavior is captured through a-priori equilibrated stress resultants. To eliminate the spurious mechanisms associated with drilling rotation on arbitrary polygons and to recover the correct rotational field, the in-plane edge field is enriched with a cubic interpolation of the normal displacement. This interpolation is based on an exact evaluation of the element-average rigid-body rotation using only vertex data, ensuring a consistent reconstruction of the drilling-related rotation. The out-of-plane edge displacement is interpolated using a cubic linked interpolation previously proposed for plates HVE, ensuring high accuracy and a locking-free behaviour. Stability analyses on both regular and highly distorted polygonal meshes confirm the robustness of the proposed formulation, demonstrating the absence of zero-energy modes. The element reproduces constant stress states exactly, even in highly distorted polygonal patch tests. Benchmark problems highlight the locking-free performance of the proposed HVE, its high accuracy even on coarse meshes, and its insensitivity to mesh distortion.