A search is presented for single production of a vector-like $B$ quark that decays into a Higgs boson and a $b$ quark. The search uses a dataset corresponding to 79.8 fb$^{-1}$ of $pp$ collisions at $\sqrt{s}=13$ TeV collected during 2015-2017 with the ATLAS detector at the Large Hadron Collider. The analysis focuses on the presence of a Higgs boson that decays into two photons, a high-momentum $b$-jet and a forward jet, characteristic of $t$-channel production of the signal process. The main background process is non-resonant diphoton production, with additional small contributions from Standard Model processes containing $H\rightarrow\gamma\gamma$ decays. No significant excess above the background expectation is observed, and upper limits on the signal production cross section are set as a function of the $B$-quark mass. Assuming a generalized coupling $\kappa_B = 0.5$ and doublet branching ratios of 50% for $B\rightarrow Hb$ and $B\rightarrow Zb$, $B$ quarks with masses less than 1210 GeV are excluded at the 95% confidence level.

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