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Active matter taps into external energy sources to power its own processes. Systems of passive particles ordinarily lack this capacity, but can become active if the constituent particles interact with each other nonreciprocally. By reformulating the theory of classical wave-matter interactions, we demonstrate that interactions mediated by scattered waves generally are not constrained by Newton's third law. The resulting center-of-mass forces propel clusters of scatterers, enabling them to extract energy from the wave and rendering them active. This form of activity is an emergent property of the scatterers' state of organization and can arise in any system where mobile objects scatter waves. Emergent activity flips the script on conventional active matter whose nonreciprocity emerges from its activity, and not the other way around. We combine theory, experiment, and simulation to illustrate how emergent activity arises in wave-matter composite systems and to explore the phenomenology of emergent activity in experimentally accessible models. These preliminary studies suggest that heterogeneity is a singular perturbation to the dynamics of wave-matter composite systems, and induces emergent activity under all but the most limited circumstances.