The paper presents a measurement of the Standard Model Higgs Boson decaying to $b$-quark pairs in the vector boson fusion (VBF) production mode. A sample corresponding to 126 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of $s=$ 13 TeV proton-proton collision data, collected with the ATLAS experiment at the Large Hadron Collider, is analyzed utilizing an adversarial neural network for event classification. The signal strength, defined as the ratio of the measured signal yield to that predicted by the Standard Model for VBF Higgs production, is measured to be $0.95-0.36+0.38$, corresponding to an observed (expected) significance of 2.6 (2.8) standard deviations from the background only hypothesis. The results are additionally combined with an analysis of Higgs bosons decaying to $b$-quarks, produced via VBF in association with a photon.