SPIRou is a near-infrared spectropolarimeter and a high-precision velocimeter. The SPIRou Legacy Survey collected data from February 2019 to June 2022, half of the time devoted to a blind search for exoplanets around nearby cool stars. The aim of this paper is to present this program and an overview of its properties, and to revisit the radial velocity (RV) data of two multiplanet systems, including new visits with SPIRou. From SPIRou data, we can extract precise RVs using efficient telluric correction and line-by-line measurement techniques, and we can reconstruct stellar magnetic fields from the collection of polarized spectra using the Zeeman- Doppler imaging method. The stellar sample of our blind search in the solar neighborhood, the observing strategy, the RV noise estimates, chromatic behavior, and current limitations of SPIRou RV measurements on bright M dwarfs are described. In addition, SPIRou data over a 2.5-year time span allow us to revisit the known multiplanet systems GJ 876 and GJ 1148. For GJ 876, the new dynamical analysis including the four planets is consistent with previous models and confirms that this system is deep in the Laplace resonance and likely chaotic. The large-scale magnetic map of GJ 876 over two consecutive observing seasons is obtained and shows a dominant dipolar field with a polar strength of 30G, which defines the magnetic environment in which the inner planet with a period of 1.94 d is embedded. For GJ 1148, we refine the known two-planet model.