Animal communication is frequently studied with conventional network representations that link pairs of individuals who interact, for example, through vocalisation. However, acoustic signals often have multiple simultaneous receivers, or receivers integrate information from multiple signallers, meaning these interactions are not dyadic. Additionally, non-dyadic social structures often shape an individual's behavioural response to vocal communication. Recently, major advances have been made in the study of these non-dyadic, higher-order networks (e.g., hypergraphs and simplicial complexes). Here, we show how these approaches can provide new insights into vocal communication through three case studies that illustrate how higher-order network models can: a) alter predictions made about the outcome of vocally-coordinated group departures; b) generate different patterns of song synchronisation than models that only include dyadic interactions; and c) inform models of cultural evolution of vocal communication. Together, our examples highlight the potential power of higher-order networks to study animal vocal communication. We then build on our case studies to identify key challenges in applying higher-order network approaches in this context and outline important research questions these techniques could help answer.