We present a numerical analysis of the vacuum structure of periodic QED in 2+1 dimensions. We use a Hartree-Fock ansatz of a bilinear Gaussian wave functional on which we impose the conditions of gauge invariance and compactness. This ansatz has the correct structure in the weak-coupling limit and can be solved by employing a dilute-gas expansion. A numerical investigation substantiates the result of an exponentially increasing dynamical mass but deviates in detail from the analytic approximation. Using a Monte Carlo program we obtain agreement with the dilute-gas approximation for weak coupling, exhibit the behavior for strong coupling, and establish the viability of this form of analysis.