We have performed studies of the 3D random field $XY$ model on $L \times L \times L$ simple cubic lattices with periodic boundary conditions, with a random field strength of $h_r = 1.875$, for $L = 64$, $96$ and $128$, using a parallelized Monte Carlo algorithm. We present results for the angle-averaged magnetic structure factor, $S(k)$ at $T = 1.00$, which appears to be the temperature at which small jumps in the magnetization per spin and the energy per spin occur. The magnetization jump per spin scales with size roughly as $L^{-3/4}$, while the energy jump per spin scales like $L^{-3/2}$. The results also indicate the existence of an approximately logarithmic divergence of $S(k)$ as $k \to 0$. The magnetic susceptibility,$\chi(\vec{\mathbf{k}} = 0)$, on the other hand, seems to have a value of about $14.2$ under these conditions. This suggests the absence of a ferromagnetic phase, and that the lower critical dimension for long-range order in this model is three. Similar results are found for $L = 64$ samples at $h_r = 2.0$ and $T = 0.875$. We expect that the behavior is qualitatively similar along the entire phase boundary. These results appear to be related to recent work on quantum disorder.