Adsorption of silver nanoparticles on polymers may affect the processes in which they participate, adjusting the analytical characteristics of methods for the quantitation of various substances. In the present study, a composite material based on silver triangular nanoplates (AgTNPs) and polyurethane foam was proposed for chemical analysis. The prospects of its application for the solid-phase/colorimetric determination of organic thiols were substantiated. It was found that aggregation of AgTNPs upon the action of thiols is manifested by a decrease in the AgTNPs’ localized surface plasmon resonance band and its significant broadening. Spectral changes accompanying the process can be registered using household color-recording devices and even visually. Four thiols differing in their functional groups were tested. It was found that their limits of detection increase in the series cysteamine < 2-mercaptoethanol < cysteine = 3-mercaptopropionic acid and come to 50, 160, 500, and 500 nM, respectively. The applicability of the developed approach was demonstrated during the analysis of pharmaceuticals and food products.