Topological defects are inherently stable structures that manifest in a variety of physical settings, from particle physics and cosmology to superfluids and quantum magnets. The geometric structure of Bloch wave functions in a periodic lattice may host topological defects, underpinning the unique properties of topological quantum matter. While pointlike defects, the celebrated Weyl points, have been extensively studied, higher dimensional structures have proven to be harder to pin down. Here, we report the experimental discovery of orbital vortex lines - the first imaging of non-trivial quantum-phase winding at line nodes - in the three-dimensional band structure of a topological semimetal, TaAs. Leveraging dichroic photoemission tomography, we directly image the winding of atomic orbital angular momentum, thereby revealing - and determining the location of - lines of vorticity in full 3D momentum space. We determine the core of the orbital angular momentum vortex to host a so-called almost movable, two-fold spin-degenerate Weyl nodal line, a topological feature predicted to occur in certain non-symmorphic crystals. These results establish the capacity to detect complex topological textures in reciprocal space and may pave the way toward novel orbital transport phenomena in metallic quantum materials.