The chiral vortical effect (CVE) – an axial current driven by rotation in chiral matter – appears in systems ranging from relativistic fluids to Weyl semimetals. We present a quantum treatment of a rotating Weyl fermion in a finite cylinder that reproduces the on-axis current density obtained by Vilenkin in an infinite system, while revealing two new phenomena. First, we find that the cross-sectional averaged current density vanishes in the thermodynamic limit, establishing that the bulk vortical response is purely a magnetization current. Second, for spin-polarized boundary conditions, we uncover a robust family of chiral modes that transport axial charge, $\Delta Q=\chi N^2\Delta \theta /4\pi$, under rotation by angle $\Delta \theta$, where $\chi$ is the Weyl node chirality and $N$ is the number of chiral modes. The pump is independent of temperature, Fermi level, and Weyl velocities, but depends on the UV-sensitive number $N$. These results clarify the quantum structure of the bulk CVE and reveal a boundary-enforced chiral spectral structure underlying vortical response in Weyl systems.