This paper presents an effective variational formalism of construction of the thick anisotropic linear shell theory of Vekua-type of an arbitrary order. The dynamic equations are formulated as the Lagrange equations of the second kind, independent from expansion functions. The asymmetric stress tensor is used to obtain the compact form of the dynamic equations similar to the classical shell theories. The proposed approach allows one for computer-aided derivation of all equations using computer algebra software supporting main tensor operations. Some solutions of test problems are presented.