We present results for Vela C obtained during the 2012 flight of the Balloon-borne Large Aperture Submillimeter Telescope for Polarimetry. We mapped polarized intensity across almost the entire extent of this giant molecular cloud, in bands centered at 250, 350, and 500 μm. In this initial paper, we show our 500 μm data smoothed to a resolution of 2.‧5 (approximately 0.5 pc). We show that the mean level of the fractional polarization p and most of its spatial variations can be accounted for using an empirical three-parameter power-law fit, p \propto {{\boldsymbol{N}}}-0.45 {{\boldsymbol{S}}}-0.60, where N is the hydrogen column density and S is the polarization-angle dispersion on 0.5 pc scales. The decrease of p with increasing S is expected because changes in the magnetic field direction within the cloud volume sampled by each measurement will lead to cancellation of polarization signals. The decrease of p with increasing N might be caused by the same effect, if magnetic field disorder increases for high column density sightlines. Alternatively, the intrinsic polarization efficiency of the dust grain population might be lower for material along higher density sightlines. We find no significant correlation between N and S. Comparison of observed submillimeter polarization maps with synthetic polarization maps derived from numerical simulations provides a promising method for testing star formation theories. Realistic simulations should allow for the possibility of variable intrinsic polarization efficiency. The measured levels of correlation among p, N, and S provide points of comparison between observations and simulations.