The associated production of Higgs and W bosons via vector-boson fusion is highly sensitive to the relative sign of the Higgs boson couplings to W and Z bosons. In this Letter, two searches for this process are presented, using 140  fb^{-1} of proton-proton collision data at sqrt[s]=13  TeV recorded by the ATLAS detector at the LHC. The first search targets scenarios with opposite-sign couplings of the W and Z bosons to the Higgs boson, while the second targets standard model-like scenarios with same-sign couplings. Both analyses consider Higgs boson decays into a pair of b quarks and W boson decays with an electron or muon. The data exclude the opposite-sign coupling hypothesis with a significance beyond 5σ, and the observed (expected) upper limit set on the cross section for vector-boson fusion WH production is 9.0 (8.7) times the standard model value at 95% confidence level.