"Spin ice" materials are described by a lattice version of electrodynamics, with elementary excitations that resemble magnetic monopoles. At low temperatures, the response of quantum spin ice is governed by the coherent quantum dynamics of monopoles. Monopoles move by flipping physical spins; the matrix element for a spin flip depends strongly on the configuration of the other spins in the vicinity of a monopole. Thus, monopole motion is "kinetically constrained." Using numerical simulations and an approximate mapping to quantum percolation, we explore the properties of monopole wavefunctions, which we find to be near a quantum percolation threshold, and discuss the implications for experimental measurements of spin response.