Vortices arise as topological excitations in superfluids and  ultracold quantum gases. Their fully nonperturbative quantization beyond the semiclassical approximation is a nontrivial issue.  Upon dualization, the (2+1)-d O(2) model turns into scalar QED, with the vortex being described by a scalar field. The massless Goldstone boson of the spontaneously broken O(2) symmetry then manifests itself as a dual photon. The vortex is a nonlocal infraparticle because it is surrounded by a cloud of massless photons. Its mass is logarithmically infrared divergent because in (2+1)-d the Coulomb potential is logarithmically confining. The Gauss law forbids the existence of a single vortex in a periodic volume. C-periodic boundary conditions introduce a charge conjugation twist and allow the existence of a single vortex in a finite volume without breaking translation invariance. Using numerical simulations, the universal finite-volume vortex mass and vortex charge are then computed accurately in the vicinity of the Wilson-Fisher fixed point.