Multiple vertebrate embryonic structures such as organ primordia are composed of a volume of confluent cells. Although mechanisms that shape tissue sheets are increasingly understood, those which shape a volume of cells remain obscure. Here we show 3D mesenchymal cell intercalations, rather than cell divisions and biophysical tissue properties, are essential to shape the mandibular arch of the mouse embryo.Using a genetically encoded vinculin tension sensor, we show that cortical force oscillations promote these intercalations. Genetic loss and gain of function approaches show that Wnt5a functions as a spatial cue to coordinate cell polarity with cytoskeletal oscillation. YAP/TAZ and PIEZO1 serve as downstream effectors of Wnt5a-mediated actomyosin bias and cytosolic calcium transients, respectively, to ensure appropriate tissue form during growth. Our data support oriented 3D cell neighbour exchange as a conserved mechanism driving volumetric morphogenesis.