Reactive methods for motion planning offer robustness advantages in the presence of large disturbances for robotic systems. Unfortunately, it is difficult to generalize these reactive methods to underactuated systems and existing research predominantly focuses on solutions based on the high-gain tracking of dynamically feasible trajectories. Self-balancing mobile robots with spherical wheels, BallBot platforms, are rich examples of such underactuated mechanisms where motion planning has traditionally been done through trajectory tracking on actuated system states with little explicit feedback on external states that are not directly controlled. In this paper, we propose a reactive path-following controller for external states of such platforms, eliminating the need to follow time-parameterized state trajectories. We first define the path-following problem in general, and present how it can be realized through a parallel composition of existing, two-dimensional controllers for the BallBot morphology. We then show in simulation how this idea can be used to achieve asymptotic convergence to a linear path with a constant forward velocity. We also show how the basic idea can be generalized to more complex path shapes such as circles, towards an eventual deployment in a more complete motion planning framework based on sequential composition.