A search for $W\gamma$ resonances in the mass range between 0.7 and 6.0 TeV is presented. The $W$ boson is reconstructed via its hadronic decays, with the final-state products forming a single large-radius jet, owing to a high Lorentz boost of the $W$ boson. The search is based on proton-proton collision data at $\sqrt{s} = 13 ~\text{TeV}$, corresponding to an integrated luminosity of 137 $\text{fb}^{-1}$, collected with the CMS detector at the LHC in 2016--2018. The $W\gamma$ mass spectrum is parameterized with a smoothly falling background function and examined for the presence of resonance-like signals. No significant excess above the predicted background is observed. Model-specific upper limits at 95% confidence level on the product of the cross section and branching fraction to the $W\gamma$ channel are set. Limits for narrow resonances and for resonances with an intrinsic width equal to 5% of their mass, for spin-0 and spin-1 hypotheses, range between 0.17 fb at 6.0 TeV and 55 fb at 0.7 TeV. These are the most restrictive limits to date on the existence of such resonances. In specific narrow-resonance benchmark models, heavy scalar (vector) triplet resonances with masses between 0.75 (1.15) and 1.40 (1.36) TeV are excluded for a range of model parameters. Model-independent limits on the product of the cross section, signal acceptance, and branching fraction to the $W\gamma$ channel are set for minimum $W\gamma$ mass thresholds between 1.5 and 8.0 TeV.