The $A_i$'s are independent zero-mean random vectors in $\mathbb{R}^{d \times d}$, which is a Hilbert space with respect to the Frobenius norm $\|\cdot\|:=\|\cdot\|_F$. So, by a vector version of Rosenthal's inequality (see e.g. [Theorem 5.2][1]), for some real universal constant $K$, 
\begin{equation}
	E\Big\|\sum_i A_i\Big\|^4\le K\Big[\sum_iE\|A_i\|^4+\Big(\sum_iE\|A_i\|^2\Big)^2\Big]. 
\end{equation}
In the iid case, we have 
\begin{align}
	E\Big\|\sum_i A_i\Big\|^4&\le K[nE\|A_1\|^4+n^2(E\|A_1\|^2)^2] \\ 
	&\le K(n+n^2)E\|A_1\|^4\le 2Kn^2E\|A_1\|^4. 
\end{align} 

[1]: https://projecteuclid.org/download/pdf_1/euclid.aop/1176988477