We present a comprehensive approach for tailoring the spectral and angular properties of infrared thermal radiation by using a polymer resonator with molecular vibrational modes, consisting of a polymer thin film on a back-reflective substrate. To precisely design the resonator, we derived the infrared dielectric function of a poly(methyl methacrylate) (PMMA) thin film from the measured reflectance spectrum by fitting it with a Gaussian-convoluted Drude-Lorentz model while accounting for the inhomogeneous broadening caused by the disordered structure of polymers. Our experimental and numerical characterization confirms that the polymer resonator exhibits spectral shaping from quasi-broadband to narrowband due to the intrinsic molecular vibrational absorption of the polymer. The frequency-isolated and strong molecular vibrational absorption of the carbonyl stretching mode at 1730 cm-1 enables the narrowband shaping of the PMMA resonator. In addition, we confirm that the angular-shaping characteristics of this polymer resonator can be tuned, from omnidirectional to strongly angular selective, by changing its polymer film thickness. Modal dispersion analysis reveals that the angle-selectivity of the polymer resonator at an angle of incidence of 80° comes from coupling between the molecular vibrational mode and leaky mode. The proposed infrared radiation management strategy based on molecular vibrational modes of polymers is cost-effective, scalable, and works well with terrestrial matter, including organic compounds and gas molecules, showing promise for applications such as optical gas sensing and radiative thermal management.