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In this paper, we investigate the existence and uniqueness of a solution for a linear Volterra-Stieltjes integral equation of the second kind, as well as for a homogeneous and a nonhomogeneous linear dynamic equations on time scales, whose integral forms contain Perron Δ-integrals defined in Banach spaces. We also provide a variation-of-constant formula for a nonhomogeneous linear dynamic equations on time scales and we establish results on controllability for linear dynamic equations. Since we work in the framework of Perron Δ-integrals, we can handle functions not only having many discontinuities, but also being highly oscillating. Our results require weaker conditions than those in the literature. We include some examples to illustrate our main results.