Thermal dark matter models with particle χ masses below the electroweak scale can provide an explanation for the observed relic dark matter density. This would imply the existence of a new feeble interaction between the dark and ordinary matter. We report on a new search for the sub-GeV χ production through the interaction mediated by a new vector boson, called the dark photon A^{'}, in collisions of 100 GeV electrons with the active target of the NA64 experiment at the CERN SPS. With 9.37×10^{11} electrons on target collected during 2016-2022 runs NA64 probes for the first time the well-motivated region of parameter space of benchmark thermal scalar and fermionic dark matter models. No evidence for dark matter production has been found. This allows us to set the most sensitive limits on the A^{'} couplings to photons for masses m_{A^{'}}≲0.35 GeV, and to exclude scalar and Majorana dark matter with the χ-A^{'} coupling α_{D}≤0.1 for masses 0.001≲m_{χ}≲0.1 GeV and 3m_{χ}≤m_{A^{'}}.