I know that there is a CLT for $\mathbb R$-valued martingale difference process that goes roughly as follows:
Let $X$ be an $\mathbb F$-martingale difference process, i.e. $\mathbf E [X_t \mid \mathcal F_{t-1}]=0$, and suppose $X$ satisfies some kind of Lindeberg condition, then $$ \frac{\sum_{i=1}^n X_i}{\sqrt{\sum_{i=1}^n \mathbf E[X_i^2]}} \xrightarrow{\mathcal D} \mathcal N_{0,1}. $$
I am searching for some multi-dimensional version of this theorem, namely when $X_t$ takes values in $\mathbb R^d$. I googled 'multivariate martingale CLT' and 'multidimensional martingale CLT' and what I found are only some obscure continuous-time results, e.g. This paper.
Is there some discrete-time multi-dimensional martingale CLT theorem that looks close to the one described above?
Use the Cramer-Wold device. That is, if $t^{\top}S_n\xrightarrow{d}t^{\top}S$ for all $t\in \mathbb{R}^d$, then $S_n\xrightarrow{d}S$, where $S_n:=\sum_{i=1}^n X_i/\sigma_n$ and $\{\sigma_n\}$ is a normalizing sequence. Note that in your case $\{t^{\top} X_n\}$ is a martingale difference sequence w.r.t. $\{\mathcal{F}_n\}$.