This article reports on a search for dijet resonances using $132$ fb$^{-1}$ of $pp$ collision data recorded at $\sqrt{s} = 13$ TeV by the ATLAS detector at the Large Hadron Collider. The search is performed solely on jets reconstructed within the ATLAS trigger to overcome bandwidth limitations imposed on conventional single-jet triggers, which would otherwise reject data from decays of TeV dijet resonances. Collision events with two jets satisfying transverse momentum thresholds of $p_{\textrm{T}} \ge 85$ GeV and jet rapidity separation of $|y^{*}|<0.6$ are analysed for dijet resonances with invariant masses from $375$ to $1800$ GeV. A data-driven background estimate is used to model the dijet mass distribution from multijet processes. No significant excess above the expected background is observed. Upper limits are set at $95\%$ confidence level on coupling values for a benchmark leptophobic axial-vector $Z^{\prime}$ model and on the production cross-section for a new resonance contributing a Gaussian-distributed line-shape to the dijet mass distribution.