We report on a search for the standard model Higgs boson produced in association with a vector boson in the full data set of proton-antiproton collisions at sqrt[s]=1.96 TeV recorded by the CDF II detector at the Tevatron, corresponding to an integrated luminosity of 9.45 fb(-1). We consider events having no identified charged lepton, a transverse energy imbalance, and two or three jets, of which at least one is consistent with originating from the decay of a b quark. We place 95% credibility level upper limits on the production cross section times standard model branching fraction for several mass hypotheses between 90 and 150 GeV/c(2). For a Higgs boson mass of 125 GeV/c(2), the observed (expected) limit is 6.7 (3.6) times the standard model prediction.