We study a two-dimensional tight-binding lattice for excitons with on-site disorder, coupled to a thermal environment at infinite temperature. The disorder acts to localize an exciton spatially, while the environment generates dynamics which enable exploration of the lattice. Although the steady state of the system is trivially uniform, we observe a rich dynamics and uncover a dynamical phase transition in the space of temporal trajectories. This transition is identified as a localization in the dynamics generated by the bath. We explore spatial features in the dynamics and employ a generalization of the inverse participation ratio to deduce an ergodic timescale for the lattice.