$\newcommand\R{\mathbb R}\newcommand\Z{\mathbb Z}\newcommand\De\Delta$This follows almost immediately from the multidimensional local limit theorem (MLLT) proved by Meĭzler, D. G.; Parasyuk, O. S.; Rvačeva, E. L.
in the paper
"On a many dimensional local limit theorem of the theory of probability." (Russian)
Ukrain. Mat. Žurnal 1, (1949). no. 1, 9--20.
The sufficiency part of their result can be stated as follows.
Let $Y_1,Y_2,\dots$ be iid random vectors with values in $\Z^d$. Let $S$ denote the support set of (the distribution of) $Y_1$. Suppose that the lattice generated by the set $S-S$ is the entire set $\Z^d$. Let $a$ and $C$ denote, respectively, the mean and the covariance matrix of (the distribution of) $Y_1$. (Note that the condition on $S$ implies that the matrix $C$ is invertible.) Let $Z_n:=Y_1+\dots+Y_n$.
Then
\begin{equation}
P(Z_n=z)=\frac1{(2\pi n)^{d/2}\sqrt\De}\,\Big[\exp\Big\{-\frac12\,Q\Big(\frac{z-na}{\sqrt n}\Big)\Big\}+o(1)\Big] \tag{1}
\end{equation}
uniformly over all $z\in\Z^d$, where $\De$ is the determinant of $C$ and $Q$ is the quadratic form with the matrix $C^{-1}$.
The paper by Meĭzler et al is in Russian; however, I think it is not hard to read it using Google Translate, say. Unfortunately, there is a mistake in the statement of the result in that paper, where $C$ is defined, not as the covariance matrix $EY_1Y_1^T-EY_1\,EY_1^T$ of $Y_1$, but as $EY_1Y_1^T$ (as usual, we identify $\Z^p$ with the set $\Z^{p\times1}$ of all $p\times 1$ column matrices with integral entries). However, the proof actually seems to be a correct proof of the correct version of the result -- see e.g. formula (10) in that paper.
Let now $X_1,X_2,\dots$ be the iid jumps of your random walk, so that $W_n:=W(n)=X_1+\dots+X_n$. Let $L$ be the minimal lattice in $\Z^d$ such that the support of the distribution of $X_1$ is contained in $c+L$ for some $c\in\Z^d$. For this minimal lattice $L$, take indeed any $c\in\Z^d$ such that the support of the distribution of $X_1$ is contained in $c+L$. Let $(b_1,\dots,b_r)$ be any basis of the lattice $L$ (where necessarily $1\le r\le d$), so that $L=\Z b_1+\dots+\Z b_r$. For each natural $j$, let $Y_j$ the $r\times 1$ column matrix of the coordinates of the random vector $X_1-c$ with respect to the basis $(b_1,\dots,b_r)$ of $L$, so that
$$Y_j=M(X_j-c)$$
for some matrix $M$. Moreover, all values of $Y_j$ are in $\Z^r$. Furthermore,
the lattice generated by the set $S-S$ is the entire set $\Z^r$, where $S$ still denotes the support set of (the distribution of) $Y_1$.
Note also that $a=EY_j=EM(X_j-c)=-Mc$, since $EX_j=0$.
It follows by (1) that uniformly in $k=nc+l\in nc+L$ with $\|k\|_2\le\sqrt n$
\begin{align*}
P(W_n=k)&=P(W_n=Ml) \\
&=\frac1{(2\pi n)^{r/2}\sqrt\De}\,\big[\exp\big\{-\frac1{2n}\,Q(Ml+nMc)\big\}+o(1)\big] \\
&=\frac1{(2\pi n)^{r/2}\sqrt\De}\,\big[\exp\big\{-\frac1{2n}\,Q(Mk)\big\}+o(1)\big] \\
&\ge\frac K{n^{r/2}}
\end{align*}
for some positive real constant $K$ independent of $n,k$. On the other hand, $P(W_n=k)>0$ only if $k\in nc+L$. Thus,
\begin{align*}
P(W_n=k)\ge\frac K{n^{r/2}}\ge\frac K{n^{d/2}}
\end{align*}
for all $k\in\Z^d$ such that $P(W_n=k)>0$, as desired.
