We propose a computational method for verifying a state-space safety constraint of a network of interconnected dynamical systems satisfying a dissipativity property. We construct an invariant set as the sublevel set of a Lyapunov function comprised of local storage functions for each subsystem. This approach requires only knowledge of a local dissipativity property for each subsystem and the static interconnection matrix for the network, and we pose the safety verification as a sum-of-squares feasibility problem. In addition to reducing the computational burden of system design, we allow the safety constraint and initial conditions to depend on an unknown equilibrium, thus offering increased flexibility over existing techniques.