Observe that if $M\models\def\zfc{\mathrm{ZFC}}\zfc$ has nonstandard $\omega$, then $x\in M$ is externally well founded iff its rank $\rho(x)$ is a standard natural number: this follows easily by iterating the observation that if $\rho(x)=\sup\{\rho(y)+1:y\in x\}$ is infinite or nonstandard, then $\rho(y)$ is infinite or nonstandard for some $y\in x$ as well. Thus, $T$ is definitionally equivalent with a similarly defined theory where we have a predicate $\def\N{\mathbb N}\N(x)$ for standard natural numbers in place of ew. I will henceforth assume $T$ is formulated in this way.
Assuming ZFC is consistent, the theory $T$ cannot be axiomatized by finitely many schemata, as it is not recursively axiomatizable. In fact, it is $\Pi^1_1$-complete. On the one hand, given a $\Pi^1_1$ sentence of the form $\forall X\,\phi(X)$ where $\phi$ is arithmetical, we have
$$\N\models\forall X\,\phi(X)\iff T\vdash\forall X\subseteq\omega\,\phi^\N(X),$$
where $\phi^\mathbb N$ denotes the relativization of all quantifiers in $\phi$ to the $\N$ predicate (in particular, this means that the formula can only query $x\in X$ for $x\in\N$, thus the quantifier over $X\subseteq\omega$ really quantifies over subsets of $\N$ coded in the model). For the left-to-right implication, if $M$ is any model of ZFC, and $X\in M$, then $M\models\phi^\N(X)$ iff $\N\models\phi(X\cap\N)$ (identifying standard natural numbers with the corresponding elements of $M$); thus, if $\N\models\forall X\,\phi(X)$, then also $M\models\forall X\subseteq\omega\,\phi^\N(X)$. For right-to-left, if $\N\not\models\phi(X)$ for some $X\subseteq\N$, then using compactness and the consistency of ZFC, there is a recursively saturated $M\models\zfc$ that contains an $X'$ such that $X'\cap\N=X$, thus $M\models T\land\neg\phi^\N(X')$.
On the other hand, $T$ is complete wrt models $(M,\N)$ where $M$ is a countable recursively saturated model of ZFC. Such a model $M$ and its satisfaction predicate can be encoded by a subset $S\subseteq\N$, and it is easy to see that the property “$S$ is a satisfaction predicate of a recursively saturated model $M$ of ZFC that $(M,\N)\models\phi$” is arithmetical. Thus, $T$ is $\Pi^1_1$.
Nevertheless we can describe $T$ in some nontrivial ways. First, by the usual completeness theorem for $\omega$-logic, the theory of models $(M,\N)$ where $M\models\zfc$ can be axiomatized by $\let\ob\overline\zfc+\{\N(\ob n):n\in\N\}+{}$the $\omega$-rule
$$\dfrac{\phi(\ob0),\phi(\ob1),\phi(\ob2),\dots}{\forall x\,(\N(x)\to\phi(x))},$$
where $\ob n$ denotes the numeral for $n$ (the usual definition of $n$ in the language of ZFC).
In order to generalize this to $T$, we can use the idea from a comment by Ali Enayat: let $\zfc^U$ denote the extension of $\zfc$ in a language with a new predicate $U(x)$, and axioms ensuring that $U$ is an unbounded class of ordinals $\alpha$ such that $(V_\alpha,{\in})$ is an elementary substructure of the universe w.r.t. all standard formulas. (The replacement schema is not extended to the new language.) Then $M\models\zfc$ is recursively saturated iff it has nonstandard $\omega$ and it expands to a model of $\zfc^U$: on the one hand, for any ordinal $\beta\in M$, the existence of $\alpha>\beta$ such that $V_\alpha^M\prec M$ follows from recursive saturation, hence we can take for $U$ the set of all such $\alpha$. On the other hand, a recursive type can be coded in a model with nonstandard $\omega$ by some $p\in M$, $p\subseteq\omega$ using overspill; if $\alpha\in U$ is such that all parameters of the type belong to $V_\alpha$, the type is finitely satisfiable in $V_\alpha$, hence using overspill again, a nonstandardly finite part of $p$ is satisfiable, hence the original type is satisfied in $V_\alpha$, and a fortiori in $V$.
Then $T$ consists of all $U$-free sentences provable in $\zfc^U+\{\N(\ob n):n\in\N\}+\exists x\in\omega\,\neg\N(x)+{}$the $\omega$-rule.
