Physical reservoir computing has traditionally exploited the dynamics of physical systems for computation, enabling tasks such as inference, classification, and prediction. Here, we introduce a fundamentally different paradigm for exploiting physical reservoirs, termed "reservoir signal acquisition" (RSA), in which a physical reservoir serves as a dynamical measurement device rather than a computational engine. In RSA, the reservoir transforms an unknown broadband waveform into a diverse set of measurements, enabling waveform reconstruction from low-rate samples beyond the Nyquist limit of any individual acquisition channel. We show that exact reconstruction of arbitrary broadband signals is achieved when the number of measurement channels satisfies $M \geq N_R$, where $N_R$ is the undersampling ratio. Moreover, spectrally or temporally sparse signals can be recovered even when $M \ll N_R$, demonstrating a compressed-sensing capability that naturally emerges from the diversity of reservoir dynamics. We experimentally validate RSA using a silicon photonic reservoir circuit. With a data-driven calibration requiring no physical model of the device, we reconstruct radio-frequency signals up to 12.5 GHz using only low-rate analog-to-digital converters (ADCs), corresponding to four times the Nyquist frequency of each ADC. These results establish RSA as a new signal acquisition paradigm based on physical reservoirs, extending their role from computation to sub-Nyquist acquisition of broadband waveforms.