The Advanced Camera for Surveys (ACS) Virgo Cluster Survey is a large program to image 100 early-type Virgo galaxies using the F475Wand F850LP bandpasses of the Wide Field Channel of the ACS instrument on the Hubble Space Telescope (HST). The scientific goals of this survey include an exploration of the three-dimensional structure of the Virgo Cluster and a critical examination of the usefulness of the globular cluster luminosity function as a distance indicator. Both of these issues require accurate distances for the full sample of 100 program galaxies. In this paper, we describe our data reduction procedures and examine the feasibility of accurate distance measurements using the method of surface brightness fluctuations (SBF) applied to the ACS Virgo Cluster Survey F850LP imaging. The ACS exhibits significant geometrical distortions owing to its off-axis location in the HST focal plane; correcting for these distortions by resampling the pixel values onto an undistorted frame results in pixel correlations that depend on the nature of the interpolation kernel used for the resampling. This poses a major challenge for the SBF technique, which normally assumes a flat power spectrum for the noise. We investigate a number of different interpolation kernels and show through an analysis of simulated galaxy images having realistic noise properties that it is possible, depending on the kernel, to measure SBF distances using distortion-corrected ACS images without introducing significant additional error from the resampling. We conclude by showing examples of real image power spectra from our survey.