We report measurements of single- and double-spin asymmetries for W^{±} and Z/γ^{*} boson production in longitudinally polarized p+p collisions at sqrt[s]=510 GeV by the STAR experiment at RHIC. The asymmetries for W^{±} were measured as a function of the decay lepton pseudorapidity, which provides a theoretically clean probe of the proton's polarized quark distributions at the scale of the W mass. The results are compared to theoretical predictions, constrained by polarized deep inelastic scattering measurements, and show a preference for a sizable, positive up antiquark polarization in the range 0.05<x<0.2.