This answer doesn't provide the exact VC-dimension of $C$, but does provide an almost linear upper bound.

The observation that a periodic box is a union of at most $2^d$ boxes could be used get a bound of $(2d^2+o(d^2))2^d$ for the VC-dimension of non-periodic boxes (using the fact that boxes have VC-dimension $2d$). I will use a similar idea to obtain a much better bound (but still not linear in $d$, unfortunately).

**Lemma.** Suppose $H_1,\ldots,H_n$ are set systems on a set $X$, each of VC-dimension at most $k$. Suppose $H$ is one of the following two set systems:

- $H_1\vee\ldots\vee H_n:=\{S_1\cup\ldots\cup S_n:S_i\in H_i\}$, or
- $H_1\wedge\ldots\wedge H_n:=\{S_1\cap\ldots\cap S_n:S_i\in H_i\}$.

Then $H$ has VC-dimension at most $k(1+o(1))n\log_2(n)$ (where $o(1)$ depends only on $n$).

**Proof.** Let $\pi_i$ be the shatter function for $H_i$, i.e.,
$$
\pi_i(m)=\max\{|\{Y\cap S:S\in H_i\}|:Y\subseteq X,~|Y|=m\}.
$$
Let $\pi$ be the shatter function for $H$. One can show $\pi(m)\leq\pi_1(m)\cdot\ldots\cdot\pi_n(m)$ for any $m$. By the Sauer-Shelah lemma, we have $\pi_i(m)\leq (em/k)^k$ for all $i$ and $m\geq k$. So $\pi(m)\leq (em/k)^{kn}$ for any $m\geq k$. In particular, if $m\geq k$ and $(em/k)^{kn}<2^m$, then $\pi(m)<2^m$, and so the VC-dimension of $H$ is less than $m$. So we just need to optimize $m$ satisfying these inequalities. The following works and is of the form stated in the lemma:
$$
m:=kn\log_2(cn\log_2(cn))
$$
where $c=e+\log_2(e)$.

So if we use case $(1)$ and write a periodic box as a union of at most $2^d$ boxes, then we get $(2d^2+o(d^2))2^d$.

Instead, we can use case $(2)$ to get a better bound. In particular, for $i\leq d$, let $C_i$ be the set of periodic boxes of the form $I_1\times\ldots\times I_d$ such that $I_j=[0,1]$ for all $j\neq i$. Then the collection $C$ of all periodic boxes is precisely $C_1\wedge\ldots\wedge C_d$. Moreover, each $C_i$ has the same VC-dimension as the set of $1$-dimensional periodic boxes, which is $3$. So altogether, this yields:

**Corollary.** The VC-dimension of $C$ is at most $(3+o(1))d\log_2(d)$.

The precise VC-dimension of $C$ appears to still be an open problem. I only found one paper discussing it, which only conjectures that the VC-dimension is linear in $d$ (but gives no known bounds). One major defect in my argument as that geometry is only being used in dimension $1$, and then the rest is just abstract combinatorics. One would expect intersections of elements from the $C_i$'s to be much better behaved than intersections of arbitrary sets. Probably the bound in the lemma can be improved a little bit, but not to something linear.