The silicon vacancy (${\mathrm{V}}_{\mathrm{Si}}$) in $3C$-SiC is studied as a center of interest in the field of quantum technologies, modeled as an electron spin (behaving as a two-state qubit in appropriate conditions) interacting through hyperfine coupling with the SiC nuclear spin bath containing $^{29}\mathrm{Si}$ and $^{13}\mathrm{C}$ nuclei in their natural isotopic concentration. We calculate the formation energies of the neutral and charged ${\mathrm{V}}_{\mathrm{Si}}$ with ab initio methods based on the density functional theory, identifying the stability of the neutral charge state for energies close to the valence band of $3C$-SiC. In addition, magnetic properties are calculated for the ${\mathrm{V}}_{\mathrm{Si}}^{\ensuremath{-}1}$ in $3C$-SiC and for ${\mathrm{V}}_{\mathrm{Si}}^{0}$ in both cubic and hexagonal SiC polytypes. We thereon evaluate, for the defect in the cubic polytype, the free induction decay and the Hahn-echo sequence on the electron spin interacting with the nuclear spin bath, shedding light on the electron spin-echo envelope modulation phenomenon and the decoherence effect by means of the cluster correlation expansion theory. We find a nonexponential coherence decay, which is a typical feature of solid-state qubits subjected to low-frequency $1/f$-type noise from the environment.