  In this paper, we characterise compactness of finite sums of finite products
of Toeplitz operators acting on the $\mathbb{C}^{d}$-valued weighted Bergman
Space, denoted $A_{\alpha}^{p}(\mathbb{B}_{n},\mathbb{C}^{d})$. The main result
shows that a finite sum of finite product of Toeplitz operators acting on
$A_{\alpha}^{p}(\mathbb{B}_n,\mathbb{C}_d)$ is compact if and only if its
Berezin transform vanishes on the boundary of the ball.
