This article deals with the vibrational behavior of composite conical shells (CCSs) reinforced with carbon nanotubes (CNTs) resting on Winkler- and Pasternak-type foundations. A generalized version of the Ambartsumian's first-order shear deformation theory (FSDT) is here proposed to handle the vibration problems for CCSs reinforced with CNTs, resting on an elastic foundation, while considering a uniform and functionally graded (FG) distribution for the reinforcement phase throughout the shell thickness. The basic equations of the problem are determined and solved in closed form by means of the Galerkin procedure. First, we check for the reliability and accuracy of the proposed formulation with respect to the available literature. It follows a systematic investigation aimed at checking the sensitivity of the structural response to the geometry, the foundation stiffness, the type of distribution, and the volume fraction of CNTs.