We show that antiferromagnetic magnons in d-wave altermagnets lead to orbital pumping with distinctive d-wave angular dependence on the direction of the Néel vector. We find that the anisotropy is generally much larger than that of spin pumping. We elucidate that this difference originates from the direct coupling of the orbital response to the electrical quadrupole moments induced by staggered local crystal field in an altermagnet. By symmetry analysis, we show that the anisotropy does not cancel for the two opposite-handedness antiferromagnetic magnons, suggesting that our prediction can be observed even with thermally excited magnons. We further analyze the handedness dependence for the situation with coherent magnons. These results establish multipolar anisotropy of orbital pumping as a direct dynamical signature of d-wave altermagnetic order and highlight the potential of altermagnets as a platform for magnon-driven orbitronics as well as for the interplay between magnons and electrical multipoles.