This work presents the Hybrid Virtual Element Method (HVEM) ant its application to membrane and plates problems in both elastic and elasto-plastic regimes. Similar to classical Virtual Element formulations, HVEM employs a polytopal discretisation of the structural domain and a displacement field interpolated only along the element contour, without requiring an explicit representation within the element interior. Within an energy-based VE projection operation, the stress field is assumed through interpolation functions that a-priori satisfy the equilibrium equations. A suitable selection of stress bases makes stabilisation terms, typical of classical VEM, unnecessary. These features enable HVEM to be effectively applied to the linear elastic analysis of membrane and thick plates, achieving high accuracy on coarse meshes and ensuring a smooth reconstruction of all stress field components. Interestingly, the same level of accuracy is maintained even in the presence of body forces, thanks to a projection scheme that inherently enforces equilibrium in the stress field interpolation. HVEM is further extended to elasto-plasticity problems, adopting a standard strain-driven formulation and a Backward-Euler time integration scheme. The stress evaluation is performed through an element-wise return mapping algorithm, preserving the structure of the interpolated stress field and, consequently, ensuring equilibrium is satisfied at the element level. Validation against classical benchmark problems confirms the reliability of the proposed approach and demonstrates the superior accuracy of polygonal meshes over quadrilateral ones for the same number of degrees of freedom.