We present a modification of the ideal gas gauge cell (IGGC) Monte Carlo simulation method (A. V. Neimark and A. Vishnyakov, J. Chem. Phys. 122, 234108 (2005)) designed for chemical potential calculations in dense and inhomogeneous multicomponent systems. To measure the chemical potentials of individual components of an m-component system, the simulation is carried out in m+1 simulations cells: the system under study (system cell) is set in chemical equilibrium with m gauge cells, one gauge cell per component. The system cell and the gauge cells are immersed into the thermal bath of a given temperature. Each gauge cell represents a finite volume reservoir, which contains only one component of the system. The molecules transferred from the system cell to the gauge cell change their identity and behave as ideal particles. The size of the gauge cell controls the level of density fluctuations for the given component in the system. This scheme, called the mesoscopic canonical ensemble, bridges the gap between the canonical and the grand canonical ensembles, which are known to be inconsistent in small systems. The chemical potential of a system component is defined as the work of insertion of one additional molecule, and it is rigorously calculated from the multidimensional equilibrium distribution of particles between the system and gauge cells. It is shown that the IGGC method is more efficient than the Widom method, especially in dense and inhomogeneous small systems. The proposed method is verified in the case of a binary mixture of supercritical methane and subcritical pentane, which features a liquid to vapor transition as the methane concentration increases. A special attention is paid to extreme confinements of several molecular diameters in the cross-section, where the inconsistency between the canonical ensemble and the grand canonical ensemble is most pronounced. The multicomponent IGGC method is illustrated drawing on the example of mixture of strongly and weakly adsorbing components confined to spherical pores with attractive walls.