Type Iax supernovae (SNe Iax) are thermonuclear explosions that are related to SNe Ia, but are physically distinct. The most important differences are that SNe Iax have significantly lower luminosity (1% - 50% that of typical SNe Ia), lower ejecta mass (~0.1 - 0.5 M_sun), and may leave a bound remnant. The most extreme SN Iax is SN 2008ha, which peaked at M_V = -14.2 mag, about 5 mag below that of typical SNe Ia. Here, we present Hubble Space Telescope (HST) images of UGC 12682, the host galaxy of SN 2008ha, taken 4.1 years after the peak brightness of SN 2008ha. In these deep, high-resolution images, we detect a source coincident (0.86 HST pixels; 0.043"; 1.1 sigma) with the position of SN 2008ha with M_F814W = -5.4 mag. We determine that this source is unlikely to be a chance coincidence, but that scenario cannot be completely ruled out. If this source is directly related to SN 2008ha, it is either the luminous bound remnant of the progenitor white dwarf or its companion star. The source is consistent with being an evolved >3 M_sun initial mass star, and is significantly redder than the SN Iax 2012Z progenitor system, the first detected progenitor system for a thermonuclear SN. If this source is the companion star for SN 2008ha, there is a diversity in SN Iax progenitor systems, perhaps related to the diversity in SN Iax explosions. If the source is the bound remnant of the white dwarf, it must have expanded significantly. Regardless of the nature of this source, we constrain the progenitor system of SN 2008ha to have an age of <80 Myr.