It is well known that competing phases can manifest within the core regions of superconducting vortices. In strong magnetic fields, neighboring vortices can overlap to give rise to long-ranged correlations in the competing phase. We show that the resulting textures can have topological character with similarities to skyrmion crystals in magnets. We illustrate this using the SO(3) theory of competing orders wherein a scalar charge-density-wave (CDW) order competes with superconductivity. We study this theory on a spherical surface enclosing a magnetic monopole, giving rise to two vortices on the surface. The lowest-energy solution has both vortex cores ordered in the same sense, with coherent CDW order developing over the entire sphere. This changes dramatically when a perturbation that breaks inversion symmetry is introduced. The CDW order becomes anticorrelated between vortices, resulting in a topologically stable configuration that is analogous to a skyrmion in a ferromagnet. We extend this idea to vortex lattices in the two-dimensional plane, modeled by the attractive Hubbard model with an orbital field. Upon introducing Rashba spin-orbit coupling (RSOC), we find spatially modulated textures with CDW order developing around vortex cores in a stripe-like anticorrelated fashion. In a wide parameter regime, RSOC lowers the energy cost for domain walls separating vortices with opposite CDW orders. As a consequence, the competing CDW order loses its rigidity and becomes short-ranged. We discuss implications for the underdoped cuprates where charge order develops without a sharp diffraction peak. © 2019 American Physical Society.