Yes, it is possible. Below I will prove two specific results in this direction: 1. If there is an infinite Dedekind-finite set, then there is such a set $A$ for $\kappa=\aleph_\omega$, 2. It is consistent with ZF (EDIT: at least assuming consistency of an inaccessible) that there is such a set $A$ for $\kappa=\aleph_{\omega_1}$, but there are no infinite Dedekind finite sets.

Both rely on the following general result: suppose there is a regular cardinal $\lambda$ and a set $B$ such that $B$ surjects onto $\lambda$, but $\lambda$ doesn't inject into $B$. Then for any cardinal $\kappa$ of cofinality $\lambda$ there is a set $A$ such that all cardinals $<\kappa$ embed into $A$, but $\kappa$ doesn't.

Indeed, let $f:B\to\lambda$ be a surjection. Let $\kappa_\alpha,\alpha<\lambda$ be an increasing sequence converging to $\kappa$. Let $A$ be the union of sets $\{b\}\times\kappa_{f(b)}$ for all $b\in B$. Clearly all $\kappa_\alpha$, and hence all cardinals below $\kappa$, embed into $A$. On the other hand, suppose we had an injection $\kappa\to A$. Composing with the obvious projection $A\to B$ gives us a well-orderable subset of $B$, which by assumption has size smaller than $\lambda$. But then we see the image of $\kappa$ is a union of less than $\lambda$ sets of size strictly smaller than $\kappa$ (since $\{b\}\times\kappa_{f(b)}$ are all strictly smaller than $\kappa$), contradicting the assumption that $\kappa$ has cofinality $\lambda$. This concludes the proof.

Now let me justify the claims from the start. If there is an infinite Dedekind-finite set $D$, then the conditions are satisfied for $\lambda=\omega$ if we take $B$ to be the set of all finite sequences of elements of $D$. It is not hard to see $B$ is infinite and Dedekind-finite if $D$ is, and $f:B\to\omega$ taking any sequence to its length is surjective. Therefore we may take $\kappa=\aleph_\omega$.

For an example not using Dedekind-finite sets, we recall that, assuming consistency of an inaccessible, it is consistent with ZF that DC holds and $\omega_1$ doesn't embed inside $\mathbb R$ (for instance, models of ZF+DC+"all sets of reals are Lebesgue measurable" satisfy this). DC (or just countable choice) implies there are no infinite Dedekind-finite sets and that $\omega_1$ is a regular cardinal. Since $\mathbb R$ surjects onto $\omega_1$ (interpret a real as a binary sequence coding some relation on $\omega$; if it codes well-order map it to its length, otherwise map to $0$), we can apply the above construction with $\lambda=\omega_1,B=\mathbb R$ and $\kappa=\aleph_{\omega_1}$.

Some interesting questions still remain, for instance: is it possible that the above holds for some $\kappa$ of countable cofinality, but there is no infinite Dedekind-finite sets? Is it possible for this to hold for *all* limit cardinals $\kappa>\omega$ (with or without an infinite Dedekind-finite set)?