  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 $150 \mathrm{GeV}/c^2$. For a Higgs
boson mass of $125 \mathrm{GeV}/c^2$, the observed (expected) limit is 6.7
(3.6) times the standard model prediction.
