In this paper, we study a broad class of McKean-Vlasov stochastic variational inequalities (MVSVIs), where both the drift coefficient $b$ and the diffusion coefficient $σ$ depend on time $t$, the state $X_t$ and its distribution $μ_t$. We establish the strong well-posedness, when $b$ is superlinear growth and locally Lipschitz continuous, and $σ$ is locally Hölder continuous, both with respect to $X_t$ and $μ_t$. Additionally, we present the first propagation of chaos result for MVSVIs.