Searches for the exclusive decays of the Higgs boson to an ω meson and a photon or a K⁎ meson and a photon can probe flavour-conserving and flavour-violating Higgs boson couplings to light quarks, respectively. Searches for these decays, along with the analogous Z boson decay to an ω meson and a photon, are performed with a pp collision data sample corresponding to integrated luminosities of up to 134 fb⁻¹ collected at s=13 TeV with the ATLAS detector at the CERN Large Hadron Collider. The obtained 95% confidence-level upper limits on the respective branching fractions are B(H→ωγ)<5.5×10−4, B(H→K⁎γ)<2.2×10−4 and B(Z→ωγ)<3.9×10−6. The limits for H→ωγ and Z→ωγ are 370 times and 140 times the Standard Model expected values, respectively. The result for Z→ωγ corresponds to a two-orders-of-magnitude improvement over the limit obtained by the DELPHI experiment at LEP.