A search for the production of a W boson and a Higgs boson through vector boson scattering (VBS) is presented, using CMS data from proton-proton collisions at collected from 2016 to 2018. The integrated luminosity of the data sample is 138. Selected events must be consistent with the presence of two jets originating from VBS, the leptonic decay of the W boson to an electron or muon, possibly also through an intermediate τ lepton, and a Higgs boson decaying into a pair of b quarks, reconstructed as either a single merged jet or two resolved jets. A measurement of the process as predicted by the standard model (SM) is performed alongside a study of beyond-the-SM (BSM) scenarios. The SM analysis sets an observed (expected) 95% confidence level upper limit of 14.3 (9.9) on the ratio of the measured VBS WH cross section to that expected by the SM. The BSM analysis, conducted within the so-called κ framework, excludes all scenarios with that are consistent with current measurements, where and and are the and coupling modifiers, respectively. The significance of the exclusion is beyond 5 standard deviations, and it is consistent with the SM expectation of .