The nccrFOAM suite is a collection of nonlinear coupled constitutive relation (NCCR) solvers for rarefied and microscale gas flows with vibrational non-equilibrium, in conjunction with the conservation laws implemented in the foam-extend framework which is an open-source solver with a General Public License (GPL 3). nccrFOAM solvers are developed as an extension to the dbnsTurbFoam solver by implementing additional algebraic constitutive relations for the non-conserved quantities of the stress tensor and heat flux vector. In contrast to Navier-Stokes-Fourier (NSF) solvers that employ first-order constitutive relations to calculate non-conserved quantities and consequently suffer from obvious shortcomings when simulating gas flows in high non-equilibrium, the second-order NCCR framework presents a novel alternative to simulate rarefied and microscale gas flows in a better and a more intuitive manner. In addition to the solver for monoatomic gases, the solvers are for the first time implemented for diatomic and polyatomic gases with translational-rotational and vibrational degrees of freedom based on second-order constitutive models in a three-dimensional framework. Towards this, a new foam-extend library has been written based on the two-temperature formulation to handle the translational-rotational and vibrational modes. The new foam-extend solver was validated for several representative problems, and an exhaustive list of tutorials is documented. The solver will certainly benefit the rarefied and microscale gas dynamics and hypersonics communities at large interested in flows involving high degrees of rarefaction, speed, and temperature variations.