This work concerns the modeling and dynamical study of a chemo-fluidic oscillator with the ability of coupling chemical and fluidic domains. The coupling is made possible by means of stimuli-responsive (also referred to as smart) hydrogels, which are able to change their volume under small variations of special thermodynamic parameters, in a reversible and reproducible manner. The paper presents a piecewise-smooth dynamical system describing the behavior of this chemo-fluidic oscillator. Specialized path-following algorithms are applied to study the periodic response of the system under parameter variations. Our investigation reveals that the origin of the oscillatory behavior is connected to the presence of a Hopf bifurcation in the system. Furthermore, the effect of several system parameters on the frequency of oscillation is studied in detail.