We find and describe the irreducible components of the space of rational curves on moduli spaces M of rank 2, stable vector bundles, with fixed determinant of odd degree, on curves C of genus g\geq2. We prove that the maximally rationally connected quotient of such a component is either the Jacobian of the curve C, or a direct sum of two copies of the Jacobian. We show that moduli spaces of rational curves on M are in one-to-one correspondence with moduli of rank 2 vector bundles on the surface P^1\times C.