Let me show that your reflection property is relatively consistent with ZFC, using a mild consistency assumption. 

**Theorem.** If $0^\sharp$ exists, then your reflection property is true in $L$ for any $\kappa$ above the first indiscernible.

**Proof.** Assume $0^\sharp$ exists, so we have a club of order-indiscernibles in $L$, which generate all of $L$. Let us enumerate them as $\delta_\alpha$ for ordinals $\alpha$. Let $\kappa=\delta_{\omega^2+\omega}$ be the $(\omega^2+\omega)$th indiscernible. 

Suppose in $L$ some model $B$ is given. Since the indiscernibles generate all of $L$, there is some term $\tau$ such that $B=\tau(\delta_{\alpha_1},\ldots,\delta_{\alpha_k})$, for some ordinals $\alpha_1<\cdots<\alpha_k$. 

Let $A=\tau(\delta_\omega,\delta_{\omega\cdot 2},\ldots,\delta_{\omega^2})$. This has size less than $\kappa$. Since there is an elementary embedding $j:L\to L$ moving these indiscernibles up to the ones for $B$, it follows that we have a map $j:A\to B$. 

But furthermore, every $b\in B$ is the value of a term $\sigma$ using possibly additional indisernibles $b=\sigma(\delta_{\alpha_1},\ldots,\delta_{\alpha_k},\vec\delta)$. The extra indiscernibles in $\vec\delta$ might be below the $\delta_{\alpha_i}$, or between them, or on top. The point now is that there are infinitely many indiscernibles between and around the $\delta_\omega$, $\delta_{\omega\cdot 2}$, etc., so that we can extend our indiscernibles defining $A$ with extra indiscernibles, still below $\kappa$, realizing the same finite order type, and define $a=\sigma(\delta_\omega,\delta_{\omega\cdot 2},\ldots,\delta_{\omega^2},\vec{\delta^*})$. 

Since there is an elementary embedding $j:L\to L$ defined by mapping the lower indiscernibles to the upper ones, we get an elementary embedding $j:A\to B$ mapping $j:a\mapsto b$, as desired.

So I have found a $\kappa$ that realizes your property. Since the least cardinal $\kappa$ with the property is definable in $L$, it will be below the first indiscernible. And since every larger cardinal than the least one will also have the property, it follows that every cardinal $\kappa$ above the first indiscernible has the reflection property. $\Box$

The theorem thus provides an upper bound on the consistency strength of your reflection property. This $0^\sharp$ hypothesis will not be optimal, however, (unless both are inconsistent), since the reflection property is first-order expressible in $L$, and so it is consistent with $V=L$. 

Nevertheless, when we apply the reflection property to structures such as $L_\theta$ in $L$, we get an $L_\alpha$ that has a lot of elementary embeddings into $L_\theta$, which strikes me as a very $0^\sharp$-like situation. But I don't know yet how to get a lower bound strictly above ZFC.