We analyze a Floquet circuit with random Clifford gates in one and two spatial dimensions. By using random graphs and methods from percolation theory, we prove in the two-dimensional (2D) setting that some local operators grow at a ballistic rate, which implies the absence of localization. In contrast, the one-dimensional model displays a strong form of localization, characterized by the emergence of left- and right-blocking walls in random locations. We provide additional insights by complementing our analytical results with numerical simulations of operator spreading and entanglement growth, which show the absence (presence) of localization in two dimensions (one dimension). Furthermore, we unveil how the spectral form factor of the Floquet unitary in 2D circuits behaves like that of quasifree fermions with chaotic single-particle dynamics, with an exponential ramp that persists up to times scaling linearly with the size of the system. Our work sheds light on the nature of disordered Floquet Clifford dynamics and their relationship to fully chaotic quantum dynamics.