We introduce a space of distributional one-forms $Ω^1_α$ on the torus $\mathbf{T}^2$ for which holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments. We show that there exists an $Ω^1_α$-valued random variable $A$ for which Wilson loop observables of axis paths coincide in law with the corresponding observables under the Yang-Mills measure in the sense of Lévy (2003). It holds furthermore that $Ω^1_α$ embeds into the Hölder-Besov space $\mathcal{C}^{α-1}$ for all $α\in(0,1)$, so that $A$ has the correct small scale regularity expected from perturbation theory. Our method is based on a Landau-type gauge applied to lattice approximations.