Let $\mathbf{X}_1, \mathbf{X}_2, ..., \mathbf{X}_n$ be independent $d$ dimensional random vectors having $0$ mean. Let $\mathbf{S}=\sum_{i=1}^n \mathbf{X}_i$ and $\mathbf{\Sigma}$ be the invertible covariance of $\mathbf{S}$. Let $\mathbf{Z} \sim \mathcal{N}(\mathbf{0}, \mathbf{\Sigma})$.
Then for all convex $U$, $$\big| \mathbb{P}[\mathbf{S} \in U] - \mathbb{P}[\mathbf{Z} \in U] \big| \leq C d^{1/4} \gamma,$$
where $\gamma = \sum_{i=1}^n \mathbb{E}\big\Vert \mathbf{\Sigma}^{-1/2}X_i \big\Vert^3_2.$
Obviously this still applies if the $\mathbf{X}_i$ are additionally identically distributed, so that they are IID.
Does a multi-variate CLT exist for exchangeable random variables? If it helps, the Berry-Esseen bound is not required.