Using the Zwicky Transient Facility alert stream, we are conducting a large spectroscopic campaign to construct a complete, volume-limited sample of transients brighter than 20 mag, and coincident within 100″ of galaxies in the Census of the Local Universe catalog. We describe the experiment design and spectroscopic completeness from the first 16 months of operations, which have classified 754 supernovae. We present results from a systematic search for calcium-rich gap transients in the sample of 22 low-luminosity (peak absolute magnitude M > −17), hydrogen-poor events found in the experiment. We report the detection of eight new events, and constrain their volumetric rate to 15% 5% of the SN Ia rate. Combining this sample with 10 previously known events, we find a likely continuum of spectroscopic properties ranging from events with SN Ia-like features (Ca-Ia objects) to those with SN Ib/c-like features (Ca-Ib/c objects) at peak light. Within the Ca-Ib/c events, we find two populations distinguished by their red (g − r 1.5 mag) or green ( mag) colors at the r-band peak, wherein redder events show strong line blanketing features and slower light curves (similar to Ca-Ia objects), weaker He lines, and lower Ca ii/O i in the nebular phase. We find that all together the spectroscopic continuum, volumetric rates, and striking old environments are consistent with the explosive burning of He shells on low-mass white dwarfs. We suggest that Ca-Ia and red Ca-Ib/c objects arise from the double detonation of He shells, while green Ca-Ib/c objects are consistent with low-efficiency burning scenarios like detonations in low-density shells or deflagrations.