The cold dark matter plus dark energy (ΛCDM) cosmological model has been successful at reproducing the large-scale structure of the Universe. However, on length scales smaller than ~1 Mpc and halo masses smaller than ~1011 M⊙, this framework is challenged by observations that halos are less centrally dense than predicted. Self-interacting dark matter (SIDM) models offer one way to reconcile these observations with theoretical predictions without affecting large-scale structure. I use simulations from the Feedback in Realistic Environments (FIRE-2) suite of Milky-Way-mass galaxies that incorporate the effects of both baryonic and SIDM interactions on a cosmological background, to compare the shape of the main dark matter halo predicted by SIDM simulations (at interaction cross-sections σ/m of 0.1, 1, and 10 cm2 g-1) with CDM simulations using the same initial conditions. I find that in the local collision region, where the rate of SIDM interactions is 1 or more per age of the Universe, the baryonic component of the halo erases many of the differences expected between the shapes of SIDM and CDM halos. Nonetheless, similarities between the size of the galactic stellar disk and the radius of the SIDM local collision region suggest that mapping the shape transition from the flattened disk-dominated region to the more spherical halo may provide some information about σ/m. I also explore the co-evolution of the dark matter and baryonic components to determine the processes leading to the present-day shape. Finally, I assess whether differences in the structure of stellar orbits within a region similar to the solar neighborhood could be sensitive to the subtle differences between SIDM and CDM galaxies.