Rotating magnon wave packets carrying orbital moments offer a pathway to unconventional transport phenomena. Here, we investigate magnon orbital moments and the magnon orbital Nernst effect in the prototypical altermagnets RuO2 and CrSb using first-principles calculations, linear response theory, and symmetry analysis. While symmetry constraints enforce vanishing equilibrium magnon orbital moments, we find that in thermal non-equilibrium a finite and robust magnon orbital Nernst effect emerges from the anisotropic Heisenberg exchange, regardless of spin-orbit coupling. This effect is intrinsically tied to the unique exchange splitting of magnon dispersions in altermagnets and is absent in conventional antiferromagnets. Magnon orbital moment transport displays markedly reduced sensitivity to the orientation of the N & eacute;el vector, temperature gradient, and magnetic domain structure compared to the magnon spin Seebeck and spin Nernst effects, enabling its persistence even in polycrystalline samples with arbitrary domain configurations. Our results position magnon orbital transport as a promising and robust functional mechanism for orbitronic and spintronic devices, and as a potential indirect probe of altermagnetism in disordered insulating systems.