A search is presented for the production of the Standard Model Higgs boson in association with a high-energy photon. With a focus on the vector-boson fusion process and the dominant Higgs boson decay into $b$-quark pairs, the search benefits from a large reduction of multijet background compared to more inclusive searches. Results are reported from the analysis of 132 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of $pp$ collision data at $s$=13 TeV collected with the ATLAS detector at the LHC. The measured Higgs boson signal yield in this final-state signature is $1.3\pm 1.0$ times the Standard Model prediction. The observed significance of the Higgs boson signal above the background is 1.3 standard deviations, compared to an expected significance of 1.0 standard deviations.