We have measured the polarizations of J/psi and psi(2S) mesons as functions of their transverse momentum p(T) when they are produced promptly in the rapidity range |y| < 0.6 with p(T) > or =5 GeV/c. The analysis is performed using a data sample with an integrated luminosity of about 800 pb(-1) collected by the CDF II detector. For both vector mesons, we find that the polarizations become increasingly longitudinal as p(T) increases from 5 to 30 GeV/c. These results are compared to the predictions of nonrelativistic quantum chromodynamics and other contemporary models. The effective polarizations of J/psi and psi(2S) mesons from B-hadron decays are also reported.