The De Donder-Weyl (DW) Hamiltonian theory of fields treats space and time variables on equal footing. Its quantization, called precanonical quantization, leads to a hypercomplex generalization of quantum formalism to field theory as it follows from the quantization of Poisson-Gerstenhaber brackets defined on differential forms. Our recent work on precanonical quantization of general relativity is extended to the teleparallel equivalent of general relativity (TEGR) in tetrad Palatini formulation. The covariant precanonical Schr\"odinger equation for quantum TEGR and the relevant operators are constructed from the quantization of generalized Dirac brackets calculated using the constraints analysis generalized to the DW Hamiltonian theory. Our analysis of the ordering ambiguities in the precanonical Schr\"odinger equation allows us to estimate the contribution to the cosmological constant from the quantum TEGR and argue its consistency with the observed value, albeit with the current error of estimation of 13 orders of magnitude due to the theoretical uncertainties in the relation between the scale $\varkappa$ introduced by precanonical quantization and the mass gap in the pure gauge sector of QCD.