We develop a time-dependent nonequilibrium Green function (NEGF) approach to the problem of spin pumping by precessing magnetization in one of the ferromagnetic layers within F vertical bar I vertical bar F magnetic tunnel junctions (MTJs) or F vertical bar I vertical bar N semi-MTJs in the presence of intrinsic Rashba spin-orbit coupling (SOC) at the F vertical bar I interface or the extrinsic SOC in the bulk of F layers of finite thickness (F, ferromagnet; N, normal metal; I, insulating barrier). To express the time-averaged pumped charge current, or the corresponding dc voltage signal in an open circuit, we construct a novel solution to double-time-Fourier-transformed NEGF equations. The two energy arguments of NEGFs in this representation are connected by the Floquet theorem describing multiphoton emission and absorption processes. Within this fully quantum-mechanical treatment of the conduction electrons, we find that (i) only in the presence of the interfacial Rashba SOC, the nonzero dc pumping voltage V-pump in F vertical bar I vertical bar N junctions can emerge at the adiabatic level (i. e., proportional to the microwave frequency), which could explain recent experiments on microwave-driven semi-MTJs [T. Moriyama et al., Phys. Rev. Lett. 100, 067602 (2008)]; (ii) a unique signature of this charge pumping phenomenon, where the Rashba SOC within the precessing F layer participates in the pumping process, is a V-pump that changes sign as the function of the precession cone angle; (iii) unlike conventional spin pumping in MTJs in the absence of any SOC, where one emitted or absorbed microwave photon is sufficient to match the exact solution in the frame rotating with the magnetization, the presence of the Rashba SOC requires taking into account up to 10 photons in order to reach the asymptotic value of pumped charge current; (iv) the disorder within F vertical bar I vertical bar F MTJs can enhance V-pump in the quasiballistic transport regime; and (v) the extrinsic SOC in F vertical bar I vertical bar F MTJs causes spin relaxation and eventually the decay of V-pump, which becomes negligible when the ratio of F layer thickness to the spin-diffusion length is around five. Our formalism can also be applied to charge and spin propagation in any noninteracting open quantum system that is brought out of equilibrium by time-dependent periodic external fields of arbitrary strength or frequency.