Two new Ag(I) complexes with coumaric carboxylic acid derivatives have been synthesized. Structural studies of these noncrystalline complexes have been performed using a methodology that combines laboratory and synchrotron techniques, supported by density functional theory calculations. The arrangement of ligands around the Ag(I) cation has been refined using infrared, extended X-ray absorption fine structure, and X-ray absorption near edge structure spectroscopies. Different coordination modes of carboxylate ligands are observed for the studied compounds. Carboxylate bridges are characteristic for the Ag(I) complex with 4-oxo-4H-1-benzopyran-2-carboxylic acid (1), while a bidentate chelating motif was found for the complex with 2-oxo-2H-1-benzopyran-3-carboxylic acid (2). Additionally, the carbonyl oxygen atom of the coumarin ring coordinates to the silver cation in complex 2, while it is inactive in complex 1. Antimicrobial evaluation has been performed for both compounds. The complexes show activity against selected bacteria as well as Candida yeast. This activity is slightly lower for bacteria and the same or higher for Candida in relation to the reference substances: ciprofloxacin or fluconazole.