Searches for circumstellar material around Type Ia supernovae (SNe Ia) are one of the most powerful tests of the nature of SN Ia progenitors, and radio observations provide a particularly sensitive probe of this material. Here we report radio observations for SNe Ia and their lower-luminosity thermonuclear cousins. We present the largest, most sensitive, and spectroscopically diverse study of prompt (delta t <~ 1 yr) radio observations of 85 thermonuclear SNe, including 25 obtained by our team with the unprecedented depth of the Karl G. Jansky Very Large Array. With these observations, SN 2012cg joins SN 2011fe and SN 2014J as a SN Ia with remarkably deep radio limits and excellent temporal coverage (six epochs, spanning 5--216 days after explosion, yielding Mdot/v_w <~ 5 x 10^-9 M_sun/yr / (100 km/s), assuming epsilon_B = 0.1 and epsilon_e = 0.1). All observations yield non-detections, placing strong constraints on the presence of circumstellar material. We present analytical models for the temporal and spectral evolution of prompt radio emission from thermonuclear SNe as expected from interaction with either wind-stratified or uniform density media. These models allow us to constrain the progenitor mass loss rates, with limits ranging from Mdot <~ 10^-9--10^-4 M_sun/yr, assuming a wind velocity v_w=100 km/s. We compare our radio constraints with measurements of Galactic symbiotic binaries to conclude that <~10% of thermonuclear SNe have red giant companions.
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