The Zwicky Transient Facility Census of the Local Universe I: Systematic search for Calcium rich gap transients reveal three related spectroscopic sub-classes

(Abridged) Using the Zwicky Transient Facility alert stream, we are conducting a large campaign to spectroscopically classify all transients occurring in galaxies in the Census of the Local Universe (CLU) catalog. The aim of the experiment is to construct a spectroscopically complete, volume-limited sample of transients coincident within 100" of CLU galaxies out to 200 Mpc, and to a depth of 20 mag. We describe the survey design and spectroscopic completeness from the first 16 months of operations. 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 (out of 754 spectroscopically classified SNe). We report the detection of eight Calcium rich gap transients, and constrain their volumetric rate to be at least $\approx 15\pm5$% of the SN Ia rate. Combining this sample with ten events from the literature, we find a likely continuum of spectroscopic properties ranging from events with SN Ia-like features (Ca-Ia objects) to SN Ib/c-like features (Ca-Ib/c objects) at peak light. Within the Ca-Ib/c events, we find two populations of events distinguished by their red ($g - r \approx 1.5$ mag) or green ($g - r \approx 0.5$ mag) spectral colors at $r$-band peak, wherein redder events show strong line blanketing signatures, slower light curves, weaker He lines and lower [Ca II]/[O I] in the nebular phase. Together, we find that the spectroscopic continuum, volumetric rates and striking old environments are consistent with the explosive burning of He shells on low mass white dwarfs. We posit that Ca-Ia and red Ca-Ib/c objects are consistent with the double detonation of He shells with high He burning efficiency, while green Ca-Ib/c objects could arise from less efficient He burning scenarios such as detonations in low density He shells or He shell deflagrations.