We present deep X-ray limits on the presence of a pre-explosion counterpart to the low-luminosity Type Ia supernova (SN Ia) 2017ejb. SN 2017ejb was discovered in NGC 4696, a well-studied elliptical galaxy in the Centaurus cluster with 894 ks of Chandra imaging between 14 and 3 years before SN 2017ejb was discovered. Using post-explosion photometry and spectroscopy of SN 2017ejb, we demonstrate that SN 2017ejb is most consistent with low-luminosity SNe Ia such as SN 1986G and SN 1991bg. Analyzing the location of SN 2017ejb in pre-explosion images, we do not detect a pre-explosion X-ray source. We use these data to place upper limits on the presence of any unobscured supersoft X-ray source (SSS). SSS systems are known to consist of white dwarfs accreting from a non-degenerate companion star. We rule out any source similar to known SSS systems with \(kT_{\rm eff} > 85\) eV and \(L_{\rm bol} > 4\times10^{38}~\text{erg s}^{-1}\) as well as models of stably-accreting Chandrasekhar-mass WDs with accretion rates \(\dot{M}>3\times10^{-8}~M_{\odot}~\text{yr}^{-1}\). These findings suggest that low-luminosity SNe Ia similar to SN 2017ejb explode from WDs that are low-mass, have low pre-explosion accretion rates, or accrete very soon before explosion. Based on the limits from SN 2017ejb and other nearby SNe Ia, we infer that \(<\)47% of SNe Ia explode in stably-accreting Chandrasekhar-mass SSS systems.