Polytopal methods are renowned for their versatility in handling complex geometries, offering significant advantages in problems involving irregular domains as well as adaptative refinement. In this context, we present the Neural Approximated Virtual Element Method (NAVEM), a novel approach that bridges classical polygonal discretizations with modern machine learning techniques. Indeed, NAVEM builds upon the Virtual Element Method (VEM) theory, replacing its (virtual) local basis functions with neural network-based surrogates that preserve functions key properties. By leveraging deep neural networks to approximate local VEM bases, NAVEM eliminates the need of problem-dependent projection operators and stabilization terms, traditionally required by the standard method to recover coercivity and consistency. Moreover, the usage of neural networks helps us to segregate the main computational effort needed to compute such approximations to the offline stage, aligning the NAVEM with a standard Finite Element Method on polygonal meshes in the online (assembly) phase. Specifically, we propose different neural network architectures and several training strategies, each offering varying levels of accuracy and aimed at minimizing distinct loss functions, with theoretical justifications provided for each approach. Particular attention is reserved for triangular meshes with hanging nodes, which assume a central role in many virtual element applications, such as adaptative strategies and Discrete Fracture Networks. Numerical experiments validate the viability of our procedure on different polygonal meshes and show the advantages of using this new procedure, especially when addressing highly non-linear problems. We also explore its application to non-linear elasticity.