Nonresonant Higgs boson pair production via vector boson fusion is searched for in the final state where each Higgs boson decays to a b quark-antiquark pair, and both Higgs bosons are highly Lorentz boosted so that they can be reconstructed as two large-radius jets. The data collected by the CMS experiment in 2016-2018 from proton-proton collisions at a centre-of-mass energy of 13 TeV is used, corresponding to an integrated luminosity of 138~\text{fb}^{-1}. A novel multivariate classifier based on graph neural networks, ParticleNet, is applied to identify the jets that contain the Higgs boson decay products. No excess is observed compared to the background-only expectation, and upper limits are set on the product of the nonresonant Higgs boson pair production cross section via vector boson fusion and the branching fraction into $\textrm{b}\bar{\textrm{b}}\textrm{b}\bar{\textrm{b}}$. The strength of the coupling between a pair of Higgs bosons and a pair of vector bosons relative to the standard model coupling is constrained within the range $0.6 < \kappa_{\textrm{2V}} < 1.4$ at $95\%$ confidence level when all other Higgs boson couplings are assumed equal to their standard model values.

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