Constraining Galactic Structure with the LISA White Dwarf Foreground

White dwarfs (WDs) comprise 95% of all stellar remnants, and are thus an excellent tracer of old stellar populations in the Milky Way. Current and planned telescopes are not able to directly probe the WD population in its entirety due to its inherently low luminosity. However, the Galactic population of double white dwarf binaries (DWDs) gives rise to a millihertz gravitational-wave foreground detectable by the Laser Interferometer Space Antenna (LISA). Here we show how characterizing the angular power of the WD foreground will enable probes of the Galactic structure in a novel way to determine whether the Galactic WD population traces the spatial distribution of young, bright stars, or traces a vertically heated spatial distribution associated with the Galaxy's oldest stellar populations. We do this using a binary population synthesis study that incorporates different Galactic spatial distributions for the DWD population. We find that the level of anisotropy in the WD foreground's angular power spectrum is dependent on the vertical scale height of the population, but show that multipole coefficients from the spherical harmonic decomposition must be considered individually because of LISA's angular resolution. Finally, we show that LISA can probe the vertical scale height of the Galactic WD population with an accuracy of 300 pc, using the hexadecapole moment of the WD foreground.