Non-Abelian states of matter, in which the final state depends on the order of the interchanges of two quasipar-ticles, can encode information immune from environmental noise with the potential to provide a robust platform for topological quantum computation. We demonstrate that phonons can carry non-Abelian frame charges at the band-crossing points of their frequency spectrum, and that external stimuli can drive their braiding. We present a general framework to understand the topological configurations of phonons from first-principles calculations using a topological invariant called Euler class, and provide a complete analysis of phonon braiding by combining different topological configurations. Taking a well-known dielectric material Al2O3 as a representative example, we demonstrate that electrostatic doping gives rise to phonon band inversions that can induce redistribution of the frame charges, leading to non-Abelian braiding of phonons. Our work provides a quasiparticle platform for realizable non-Abelian braiding in reciprocal space, and expands the tool set for studying braiding processes.