  We present multi-wavelength follow-up observations of the ATLAS discovered
faint Iax supernova SN 2020kyg that peaked at an absolute magnitude of $M_g
\approx -14.9 \pm 0.2$, making it another member of the faint Iax supernova
population. The bolometric light curve requires only $\approx 7 \times 10^{-3}$
M$_{\odot}$ of radioactive $^{56}$Ni, with an ejected mass of $M_{\rm ej} \sim
0.4$ M$_{\odot}$ and a low kinetic energy of $E \approx 0.05 \pm 0.02 \times
10^{51}$ erg. We construct a homogeneous volume-limited sample of 902
transients observed by ATLAS within 100 Mpc during a 3.5 year span. Using this
sample, we constrain the rates of faint Iax ($M_r \gtrsim -16$) events within
60 Mpc at $12^{+14}_{-8}\%$ of the SN Ia rate. The overall Iax rate, at
$15^{+17}_{-9}\%$ of the Ia rate, is dominated by the low-luminosity events,
with luminous SNe Iax ($M_r \lesssim -17.5$) like 2002cx and 2005hk accounting
for only $0.9^{+1.1}_{-0.5}\%$ of the Ia rate (a 2$\sigma$ upper limit of
approximately 3\%). We favour the hybrid CONe WD + He star progenitor channel
involving a failed deflagration of a near Chandrasekhar mass white dwarf,
expected to leave a bound remnant and a surviving secondary companion, as a
candidate explanation for faint Iax explosions. This scenario requires short
delay times, consistent with the observed environments of SNe Iax. Furthermore,
binary population synthesis calculations have suggested rates of $1-18\%$ of
the SN Ia rate for this channel, consistent with our rate estimates.
