A search is made for a vectorlike $T$ quark decaying into a Higgs boson and a top quark in 13 TeV proton-proton collisions using the ATLAS detector at the Large Hadron Collider with a data sample corresponding to an integrated luminosity of $139\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The Higgs-boson and top-quark candidates are identified in the all-hadronic decay mode, where $H\ensuremath{\rightarrow}b\overline{b}$ and $t\ensuremath{\rightarrow}bW\ensuremath{\rightarrow}bq{\overline{q}}^{\ensuremath{'}}$ are reconstructed as large-radius jets. The candidate Higgs boson, top quark, and associated $B$ hadrons are identified using tagging algorithms. No significant excess is observed above the background, so limits are set on the production cross section of a singlet $T$ quark at 95% confidence level, depending on the mass ${m}_{T}$ and coupling ${\ensuremath{\kappa}}_{T}$ of the vectorlike $T$ quark to Standard Model particles. In the considered mass range between 1.0 and 2.3 TeV, the upper limit on the allowed coupling values increases with ${m}_{T}$ from a minimum value of 0.35 for $1.07<{m}_{T}<1.4\text{ }\text{ }\mathrm{TeV}$ to 1.6 for ${m}_{T}=2.3\text{ }\text{ }\mathrm{TeV}$.