  Type Iax supernovae (SNe~Iax) are the most common class of peculiar SNe.
While they are thought to be thermonuclear white-dwarf (WD) SNe, SNe~Iax are
observationally similar to, but distinct from SNe~Ia. Unlike SNe~Ia, where
roughly 30\% occur in early-type galaxies, only one SN~Iax has been discovered
in an early-type galaxy, suggesting a relatively short delay time and a
distinct progenitor system. Furthermore, one SN~Iax progenitor system has been
detected in pre-explosion images with its properties consistent with either of
two models: a short-lived (<100 Myr) progenitor system consisting of a WD
primary and a He-star companion, or a singular Wolf-Rayet progenitor star.
Using deep \textit{Hubble Space Telescope} images of nine nearby SN~Iax host
galaxies, we measure the properties of stars within 200 pc of the SN position.
The ages of local stars, some of which formed with the SN progenitor system,
can constrain the time between star formation and SN, known as the delay time.
We compare the local stellar properties to synthetic photometry of
single-stellar populations, fitting to a range of possible delay times for each
SN. With this sample, we uniquely constrain the delay-time distribution for
SNe~Iax, with a median and $1-\sigma$ confidence interval delay time of $63_{-
15}^{+ 58} \times 10^{6}$ years. The measured delay-time distribution provides
an excellent constraint on the progenitor system for the class, indicating a
preference for a WD progenitor system over a Wolf-Rayet progenitor star.
