Lattices of interacting gyroscopes naturally support band gaps and topologically protected wave transport along material boundaries. Recently the authors and their collaborators found that amorphous arrangements of such coupled gyroscopes also support nontrivial topological phases. In contrast to periodic systems, for which there is a comprehensive understanding and predictive framework for band gaps and band topology, the theory of spectral gaps and topology for amorphous materials remains less developed. Here we use experiments, numerics, and analytic tools to address the relationship between local interactions and nontrivial topology. We begin with a derivation of the equations of motion within the framework of symplectic mechanics. We then present a general method for predicting whether a gap exists and for approximating the Chern number using only local features of a network, bypassing the costly diagonalization of the system's dynamical matrix. Finally we study how strong disorder interacts with band topology in gyroscopic metamaterials and find that amorphous gyroscopic Chern insulators exhibit similar critical behavior to periodic lattices. Our experiments and simulations additionally reveal a topological Anderson insulation transition, wherein disorder drives a trivial phase into a topological one.