  Type Ia supernovae (SNe Ia) produce most of the Fe-peak elements in the
Universe and therefore are a crucial ingredient in galactic chemical evolution
models. SNe Ia do not explode immediately after star formation, and the
delay-time distribution (DTD) has not been definitively determined by supernova
surveys or theoretical models. Because the DTD also affects the relationship
among age, [Fe/H], and [$\alpha$/Fe] in chemical evolution models, comparison
with observations of stars in the Milky Way is an important consistency check
for any proposed DTD. We implement several popular forms of the DTD in
combination with multiple star formation histories for the Milky Way in
multi-zone chemical evolution models which include radial stellar migration. We
compare our predicted interstellar medium abundance tracks, stellar abundance
distributions, and stellar age distributions to the final data release of the
Apache Point Observatory Galactic Evolution Experiment (APOGEE). We find that
the DTD has the largest effect on the [$\alpha$/Fe] distribution: a DTD with
more prompt SNe Ia produces a stellar abundance distribution that is skewed
toward a lower [$\alpha$/Fe] ratio. While the DTD alone cannot explain the
observed bimodality in the [$\alpha$/Fe] distribution, in combination with an
appropriate star formation history it affects the goodness of fit between the
predicted and observed high-$\alpha$ sequence. Our model results favor an
extended DTD with fewer prompt SNe Ia than the fiducial $t^{-1}$ power law.
