In type-II superconductors, magnetic fields modulate the amplitude and phase of the superconducting order parameter, forming quantized vortices where superconductivity is locally suppressed and exotic bound states or competing electronic orders emerge. Using spectroscopic-imaging scanning tunneling microscopy on epitaxial Ba(Fe_(0.94)Co_0.06)₂As₂ films, we discover an incommensurate charge-stripe order aligned with the Fe-Fe bond direction. These charge modulations become strongly enhanced at the vortex core and extend far into the vortex halo. Strikingly, the charge order modulates Andreev bound states of vortices at non-zero energies, producing abelian vortices with half-odd-integer level quantization and non-abelian vortices with integer-quantized core states that host a Majorana zero mode. The distinct vortex types are distinguished by the registry of their centers relative to the charge-stripe pattern and remain robust in ultrathin (2.5-unit-cell) films. Our findings reveal a density-wave-textured vortex matter and provide fresh insights into intertwined phenomena in iron-based superconductors.