Let me address the question in the body of the post, rather than the title question. Namely, you asked whether your theory interprets ZFC.

**Negative answer with axiom as stated.** The natural reading of your axiom $\forall\kappa\exists x\forall\alpha(\alpha\leq\kappa\to x\in\alpha)$ in set theory would be that it asserts the existence of a set $x$ containing every ordinal $\alpha$ with $\alpha\leq\kappa$, which is to say, it asserts the existence of the ordinal successor $\kappa+1$.

On this reading of your axiom, the answer is negative, the theory would not interpret ZFC. As the other answers have noted, it can build $L$, but the $L$ that it builds will not necessarily be a model of ZFC.

The reason is that the theory ZFC-P already proves the existence of ordinal successors, but it is too weak to interpret ZFC. This is a consequence of the fact that ZFC proves Con(ZFC-P), since ZFC proves that the structure of hereditarily countable sets HC is a model of ZFC-P. But ZFC-P cannot interpret a theory that proves Con(ZFC-P) by the second incompletness theorem.

More concretely, if one starts in the ZFC-P model HC of hereditarily countable sets, then the $L$ that one builds will be exactly $L_{\omega_1}$, which is a model of $V=L$, but it is not necessarily a model of ZFC.

**Positive answer with corrected axiom.** You have clarified in the comments, however, and in the post that you have an idiosyncratic meaning for $\leq$ in your axiom and that you intend it to assert the existence of cardinal successors. That is, we are working in ZFC-P + every cardinal $\kappa$ has a successor cardinal $\kappa^+$.

In this case, the answer becomes affirmative. As we noticed, ZFC-P can construct the inner model $L$, and if every cardinal has a successor, then those cardinals will arise inside $L$ and so $L$ will also think that every cardinal has a successor. And it will think that the replacement axiom is true, if replacement holds in $V$. We will get the power set axiom being true inside $L$ because the subsets of a set show up in the $L$ hierarchy before the next cardinal stage, and so we can collect them into a set by applying comprehension to that stage $L_\gamma$. In this way, all the ZFC axioms will be true in the $L$ that we build, and so we are interpreting ZFC.

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