The moiré engineering of two-dimensional magnets opens unprecedented opportunities to design novel magnetic states via the stacking-dependent magnetism. Here, we explore the formation and control of ground state topological spin structures (TSTs) in moiré CrI3 without including the nearest-neighbor (NN) Dzyaloshinskii-Moriya interactions (DMI) and dipolar interactions in the theoretical approach. Using stochastic Landau-Lifshitz-Gilbert simulations, we unveil the emergence of vortex and antivortex interlayer exchange fields at large moiré periodicity. The whirling fields stabilize spontaneous and field-assisted ground state TSTs with various topologies, including skyrmionic clusters with high topological charges. Furthermore, by examining the effect of the Kitaev interaction and the next NN DMI, we propose the latter as the unique spin-orbit coupling mechanism compatible with the experimental results on monolayer and twisted CrI3. Therefore, our study goes beyond the current knowledge about TSTs in moiré magnets, opens exciting opportunities for moiré skyrmionics, and uncovers the spin-orbit coupling in CrI3.