Two-dimensional second-order topological superconductors (SOTSCs) have gapped bulk and edge states, with zero-energy Majorana bound states localized at corners. Motivated by recent advances in Majorana nanowire experiments, we propose to realize a tunable SOTSC as a two-dimensional nanowire array. We show that the coupling between the Majorana modes of adjacent wires can be controlled by phase-biasing the device, allowing one to access a variety of topological phases. We characterize the system using scattering theory, which provides access to its transport properties and its topological invariants. The setup is robust against disorder, both in the nanowires themselves and in the Josephson junctions formed between adjacent wires. Further, we identify a parameter regime in which an initially trivial system is rendered topological upon adding disorder, providing an example of a second-order topological Anderson phase.