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Constraining Galactic Structure with the LISA White Dwarf Foreground

White dwarfs 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 white dwarf population in its entirety due to its inherently low luminosity. However, the Galactic population of double white dwarf binaries gives rise to a millihertz gravitational-wave foreground detectable by the Laser Interferometer Space Antenna (LISA). Here we show how characterizing this foreground's angular power spectrum will enable us to probe the Galactic structure in a novel way and measure the vertical scale height of the Galaxy's oldest stellar populations. We do this using a binary population synthesis study that incorporates different Galactic spatial distributions for the double white dwarf population. We find that the level of anisotropy in the white dwarf foreground's angular power spectrum is strongly dependent on the vertical scale height of the population. Finally, we show that LISA can probe the vertical scale height of the Galactic double white dwarf population with an accuracy of 50 pc-200pc, depending on angular resolution limits, using the angular power spectrum of the white dwarf foreground.