We compute radiative corrections to the superallowed \(\beta\) decay of \(^{10}\)C in an effective field theory approach using nuclear matrix elements obtained from quantum Monte Carlo calculations. These corrections are an important ingredient in the extraction of the Cabibbo-Kobayashi-Masakawa quark mixing matrix element \(V_{ud}\), and the role of this work is to illuminate the uncertainties arising from nuclear structure. Our results provide good agreement with both the traditional extraction of \(V_{ud}\), as well as with a more recent evaluation performed using the no-core shell model and a dispersion formalism. The dominant uncertainty in this approach is the presence of two unknown low-energy constants that enter into the relevant nuclear matrix elements. Future determinations of these low-energy constants -- either from QCD or modeling them with two nucleon amplitudes -- would improve the precision of the extraction in this formalism.