Shifting the focus of Type Ia supernova (SN Ia) cosmology to the near infrared (NIR) is a promising way to significantly reduce the systematic errors, as the strategy minimizes our reliance on the empirical width-luminosity relation and uncertain dust laws. Observations in the NIR are also crucial for our understanding of the origins and evolution of these events, further improving their cosmological utility. Any future experiments in the rest-frame NIR will require knowledge of the SN Ia NIR spectroscopic diversity, which is currently based on a small sample of observed spectra. Along with the accompanying paper, Phillips et al., we introduce the Carnegie Supernova Project-II (CSP-II), to follow-up nearby SNe Ia in both the optical and the NIR. In particular, this paper focuses on the CSP-II NIR spectroscopy program, describing the survey strategy, instrumental setups, data reduction, sample characteristics, and future analyses on the data set. In collaboration with the Harvard-Smithsonian Center for Astrophysics (CfA) Supernova Group, we obtained 661 NIR spectra of 157 SNe Ia. Within this sample, 451 NIR spectra of 90 SNe Ia have corresponding CSP-II follow-up light curves. Such a sample will allow detailed studies of the NIR spectroscopic properties of SNe Ia, providing a different perspective on the properties of the unburned material; the radioactive and stable nickel produced; progenitor magnetic fields; and searches for possible signatures of companion stars.