We report on the results of a search for standard model Higgs bosons produced in association with W bosons from pp[over] collisions at sqrt[s]=1.96 TeV. The search uses a data sample corresponding to approximately 1 fb(-1) of integrated luminosity. Events consistent with the W-->lnu and H-->bb[over] signature are selected by triggering on a high-p(T) electron or muon candidate and tagging one or two of the jet candidates as having originated from b quarks. A neural network filter rejects a fraction of tagged charm and light-flavor jets, increasing the b-jet purity in the sample. We observe no excess lnubb[over] production beyond the background expectation, and we set 95% confidence level upper limits on the production cross section times branching fraction sigma(pp[over]-->WH)Br(H-->bb[over]) ranging from 3.9 to 1.3 pb, for specific Higgs boson mass hypotheses in the range 110 to 150 GeV/c2, respectively.