As I understand, the axiom of infinity in ZFC theory gives us an infinite set from which the natural numbers can be extracted. (See "Axiom of infinity".)
Might the following proposed axiom be a weaker version (using everyday functional notation)?
$\exists X,S,x_0( S:X\to X\land \forall a,b\in X(S(a)=S(b)\to a=b) \land x_0\in X \land \forall a\in X(S(a)\ne x_0 ))$
In words, there exists $X,S$ and $x_0$ such that $S$ is a an injective function on $X$, and $x_0\in X$, and $x_0$ has no pre-image in $X$ under $S$.
Note: It can be shown that $X$ is Dedekind-infinite, and that it is possible to extract the natural numbers (as defined by Peano's Axioms) from $X$ where $S$ is the usual successor function and $x_0$ is the $0$ (or $1$).
Your axiom is actually equivalent to the Axiom of Infinity modulo ZFC-{Infinity}. This is very easy to see via the replacement axiom, but it follows even just using Extensionality, Pairs, Unions, Specification and Powers: you can prove a version of the recursion theorem, from which you can extract the usual set $\omega$ of von Neumann numerals.