The first search for Standard Model (SM) Higgs bosons decaying into a charm quark–antiquark pair ($H\to c\bar{c}$) via vector-boson fusion (VBF) production at the Large Hadron Collider is presented. The datasets used correspond to integrated luminosities of 37.5 and 51.5 fb${}^{-1}$ and were collected by the ATLAS detector from proton–proton collisions at $\sqrt{s}=13$ and 13.6 TeV, respectively. The observed (expected) upper limit on the $H\to c\bar{c}$ production cross-section times branching ratio is 41 (28) times the SM prediction at 95% confidence level (CL). Combining this search with a previous $H\to c\bar{c}$ search in associated production with a $W$ or $Z$ boson yields an observed (expected) limit on the Higgs–charm Yukawa coupling modifier of $|_c| < 4.7, (3.9) $.Higgsbosondecayintoabottomquarkantiquarkpair($H b{b}$)ismeasuredsimultaneouslyusingthe51.5fb$^{−1}$ dataset at $\sqrt{s}=13.6$ TeV, with an observed signal strength of 0.97(+0.57, −0.50) relative to the SM expectation. When combined with previous $H\to b\bar{b}$ results at 13 TeV, the observed (expected) significance reaches 3.2 (3.6) standard deviations, providing evidence for $H\to b\bar{b}$ events from VBF.