Establishing the physical mechanism governing exchange interactions is fundamental for exploring exotic phases such as quantum spin liquids in real materials. In this Letter, we address exchange interactions in Sr2CuTexW1-xO6, a series of double perovskites that realize a spin-1/2 square lattice and are suggested to harbor a quantum spin liquid ground state arising from the random distribution of nonmagnetic ions. Our ab initio multireference configuration interaction calculations show that replacing Te atoms with W atoms changes the dominant couplings from nearest to next-nearest neighbor due to the crucial role of unoccupied states of the nonmagnetic ions in the super-superexchange mechanism. Combined with spin-wave theory simulations, our calculated exchange couplings provide an excellent description of the inelastic neutron scattering spectra of the parent compounds, as well as explaining that the magnetic excitations in Sr2CuTe0.5W0.5O6 emerge from bond-disordered exchange couplings. Our results demonstrate the crucial role of the nonmagnetic cations in exchange interactions paving the way to further explore quantum spin liquid phases in bond-disordered materials.