This paper proposes an approach for analyzing the free vibrations of sandwich shallow shells with an auxetic core exhibiting a negative Poisson’s ratio. The face sheets are made of functionally graded materials (FGMs). The method allows to consider the panels with arbitrary planform geometries. The shell is assumed to be resting on a Pasternak elastic foundation. Mathematical formulation is based on the first order shear deformation theory (FSDT). The core’s unit cell is modeled with a hexagonal configuration, and established analytical relations are used to determine its material properties. Power law distribution is applied to characterize the effective properties of the FGM face sheets. To address shells with arbitrary planforms, the R-functions theory is integrated with the variational Ritz method. The accuracy and efficiency of the proposed methodology are demonstrated through comparison with existing results as well as new results obtained for auxetic shells of complex geometries.