Rotational transitions are unique identifiers of molecular species, including isotopologues. This article describes the rotational detections of two laser-volatilized salts, NaCl and KCl, made with a miniature Fourier transform millimeter-wave (FTmmW) cavity spectrometer that could one day be used to measure solid composition in the field or in space. The two salts are relevant targets for icy moons in the outer solar system, and in principle, other molecular solids could be analyzed with the FTmmW instrument. By coupling the spectrometer to a collisionally cooling laser ablation source, (a) we demonstrate that the FTmmW instrument is sensitive enough to detect ablation products, and (b) we use the small size of the FTmmW cavity to measure ablation product signal along the carrier gas beam. We find that for 532 nm nanosecond pulses, ablated molecules are widely dispersed in the carrier-gas jet. In addition to the miniature spectrometer results, we present several complementary measurements intended to characterize the laser ablation process. For pulse energies between 10 and 30 mJ, the ablation product yield increases linearly, reaching approximately 1012 salt molecules per 30 mJ pulse. Using mass spectrometry, we observe Li+, Na+, and K+ in the plumes of ablated NaCl, KCl, and LiCl, which implies dissociation of the volatilized material. We do not observe salt ions (e.g., NaCl+). However, with 800 nm femtosecond laser pulses, the triatomic ion clusters Li2Cl+, Na2Cl+, and K2Cl+ are produced. Finally, we observe incomplete volatilization with the nanosecond pulses: some of the ejecta are liquid droplets. The insights about ablation plume physics gleaned from these experiments should guide future implementations of the laser-volatilization technique.