Motivated by recent studies of gas permeation through polymer networks, we consider a collection of ordinary random walks of fixed length ℓ, placed randomly on the bonds of a square lattice. These walks model polymers, each with ℓ segments. Using computer simulations, we find the critical concentration of occupied bonds (i.e., the critical occupation probability) for such a network to percolate the system. Though this threshold decreases monotonically with ℓ, the critical “mass” density, defined as the total number of segments divided by total number of bonds in the system, displays a more complex behavior. In particular, for fixed mass densities, the percolation characteristics of the network can change several times, as shorter polymers are linked to form longer ones.