Chip-scale optical frequency combs can provide broadband spectroscopy for diagnosing complex organic molecules. They are also promising as miniaturized laser spectrometers in applications ranging from atmospheric chemistry to geological science and the search for extraterrestrial life. While optical cavities are commonly used to boost sensitivity, it is challenging to realize a compact cavity-enhanced comb-based spectrometer. Here, we apply the Vernier technique to free-running operation of an interband cascade laser frequency comb in a simple linear geometry that performs cavity-enhanced chemical sensing. A centimeter-scale high-finesse cavity simultaneously provides selective mode filtering and enhancement of the path length to 30 meters. As a proof-of-concept, we sense transient open-path releases of ppm-level difluoroethane with 2 ms temporal resolution over a 1 THz optical bandwidth centered at 3.64 $\mu$m.