There are a few bits of good news for an affirmative answer to this question.
Theorem 1) ZFC can be interpreted in Th(On), the first-order theory of ordinals. See Gaisi Takeuti, "Formalization of the Theory of Ordinals", JSL 1965. (https://projecteuclid.org/euclid.jsl/1183735178)
Theorem 2) There are abelian $p$-groups of every infinite ordinal length, where the length $\ell(G)$ of a group $G$ is the least ordinal $\sigma$ such that $p^\sigma G=0$. See Laszlo Fuchs, Infinite Abelian Groups, vol 2: p. 58 for the definition and p. 85 for the construction of these generalized Prufer groups.
Putting these together, I had hoped to encode the ordinals by such groups, and thus interpret Th(On) in ETCG, from which an interpretation of ZFC in ETCG would follow.
The bad news is that the theory Th(On) is a theory in a large language, which starts with $a=b$, $a<b$, $(a,b)$ (ordered pair), and then goes on to include $+$, $\times$ and all primitive recursive functions of ordinals. So to interpret Th(On) in ETCG, we would at a minimum need to find formulas $\phi_\le, \phi_{\wedge}$ in ETCG such that:
$\phi_\le (a,b)$ holds exactly when $\ell(a)\le\ell(b)$
$\phi_\wedge (a,b,c)$ holds exactly when $\ell(a)=\ell(b)^{\ell(c)}$
Perhaps $\phi_\le$ would be as simple as saying that there is a mono from $a$ to $b$. But finding $\phi_\wedge$ seems difficult.
Even finding a way of characterizing the generalized Prufer groups in the ETCG language seems difficult. Fortunately there is still one more bit of good news:
Claim 3) We can characterize the abelian groups in the language of ETCG. (I expect there are mistakes in the below, but I also expect that the mistakes can be corrected.)
$1$ is the unique terminal object in the category
a morphism is constant if it factors through $1$.
$\mathbb{Z}$ is the unique object in the category such that for every $G$ not equal to $1$, there is a non-constant map $\mathbb{Z} \rightarrow G$.
$G$ has two elements iff there are exactly two maps from $\mathbb{Z}$ to $G$.
$G$ has eight elements iff there are exactly eight maps from $\mathbb{Z}$ to $G$.
$H$ is a subgroup of $G$ iff there is a mono from $H$ to $G$.
$G/H=K$ iff there is a mono and an epi
$$H \hookrightarrow G \twoheadrightarrow K$$
whose composition is constant, and such that whenever the square commutes in the diagram below, there is a map from $\mathbb{Z}$ to $H$ making the triangle commute also:
$$\begin{array}{ccccc} & & \mathbb{Z} & \rightarrow & 1 \\ & \swarrow & \downarrow & & \downarrow \\ H & \hookrightarrow & G & \twoheadrightarrow & K\\ \end{array}$$ (The second condition is saying that the kernel of the epi is included in the range of the mono.)
$H$ is a normal subgroup of $G$ iff $G/H=K$ for some $K$.
$G$ is cyclic iff it is $\mathbb{Z}/H$ for some $H$.
$Q$ is the unique 8-element group which is not cyclic, but contains a two-element subgroup $S$ that is contained in every subgroup $R$ of $Q$.
$G$ is abelian iff all of its subgroups are normal, and $G$ is not the direct product of $Q$ with another group.
I expect we can go further towards characterizing the generalized Prufer groups by characterizing reduced non-separable infinite abelian 2-groups in ETCG. Anyone who finds that easy would be in a better place than I am to complete the difficult but maybe not impossible plan above.