A search for Higgs boson pair production via vector-boson fusion is performed in the Lorentz-boosted regime, where a Higgs boson candidate is reconstructed as a single large-radius jet, using 140 fb−1 of proton–proton collision data at s=13 TeV recorded by the ATLAS detector at the Large Hadron Collider. Only Higgs boson decays into bottom quark pairs are considered. The search is particularly sensitive to the quartic coupling between two vector bosons and two Higgs bosons relative to its Standard Model prediction, κ2V. This study constrains κ2V to 0.55<κ2V<1.49 at the 95% confidence level. The value κ2V=0 is excluded with a significance of 3.8 standard deviations with other Higgs boson couplings fixed to their Standard Model values. A search for new heavy spin-0 resonances that would mediate Higgs boson pair production via vector-boson fusion is carried out in the mass range of 1–5 TeV for the first time under several model and decay-width assumptions. No significant deviation from the Standard Model hypothesis is observed and exclusion limits at the 95% confidence level are derived.