We theoretically demonstrate that axial phonons, lattice vibration quanta carrying finite angular momentum, can host a $d$-wave angular momentum texture in orbital altermagnets in the absence of spin-orbit coupling. We consider a minimal electronic tight-binding model with $d$-wave loop-current order that breaks time-reversal symmetry. Within the Born-Oppenheimer approximation, we incorporate electron-phonon coupling via the molecular Berry curvature and show that the underlying $d$-wave orbital magnetic moment texture of the electronic state is transferred to the phonons without requiring the relativistic spin-orbit coupling. Our results expand the range of platforms available for engineering axial phonons and point to functionality unique to $d$-wave textures, including angular-momentum Seebeck and splitter effects, corresponding to longitudinal and transverse angular-momentum currents driven by a temperature gradient.