© 2020, The Author(s). A search for Higgs boson pair production via vector-boson fusion (VBF) in the bb¯ bb¯ final state is carried out with the ATLAS experiment using 126 fb−1 of proton- proton collision data delivered at s = 13 TeV by the Large Hadron Collider. This search is sensitive to VBF production of additional heavy bosons that may decay into Higgs boson pairs, and in a non-resonant topology it can constrain the quartic coupling between the Higgs bosons and vector bosons. No significant excess relative to the Standard Model expectation is observed, and limits on the production cross-section are set at the 95% confidence level for a heavy scalar resonance in the context of an extended Higgs sector, and for non-resonant Higgs boson pair production. Interpretation in terms of the coupling between a Higgs boson pair and two vector bosons is also provided: coupling values normalised to the Standard Model expectation of κ2V< −0.76 and κ2V> 2.90 are excluded at the 95% confidence level in data. [Figure not available: see fulltext.]