We perform an ab initio quantum Monte Carlo calculation of the isospin-symmetry-breaking correction \(\delta_C\) to the superallowed \(\beta\) decay of \(^{10}\)C. Using both phenomenological and chiral nuclear interactions, we evaluate the Fermi matrix element and quantify its deviation from the canonical \(\sqrt{2}\) value. The resulting \(\delta_C\) values lie in the range ≈0.15--0.25% and are consistent, within sizable uncertainties (approximately 34%--65% relative), across Hamiltonians, indicating no statistically significant dependence on the choice of nuclear interaction. The extracted values of Vud are also found to be compatible with current determinations within these uncertainties.