I think the answer is (consistently) no.

Let $\kappa$ be Mahlo in $L$ and let $\eta$ be the least ordinal such that $\kappa < \eta$ and $L_\eta \models \mathsf{ZFC}$. Consider the model $N = L_\eta[G]$ where $G \subset \text{Col}(\omega,\mathord{<}\kappa)$ is an $L_\eta$-generic filter.

Then in $N$ there is an uncountable transitive model of $\mathsf{ZFC}$, namely $L_\kappa$. It suffices to show that every such model has an inaccessible cardinal.

Let $M$ be an uncountable transitive model of $\mathsf{ZFC}$ in $N$. Note that $\text{Ord}^M \ge \kappa$ by uncountability and in fact $\text{Ord}^M = \kappa$ because otherwise $L^M$ would violate the minimality of $\eta$.

The set of $L$-inaccessibles below $\kappa$ is stationary in $L$ by definition, and this stationarity is preserved by the Levy collapse forcing that adds $G$ to obtain $N$. So the model $M$ satisfies "the class of $L$-inaccessibles is definably stationary."

Therefore (because $M \models \mathsf{ZFC}$) there is an $L$-inaccessible $\alpha < \kappa$ that is a strong limit cardinal in $M$. Then $\alpha$ is inaccessible in $M$ by Jensen's covering theorem (note that $0^\sharp$ could not have been added by forcing over $L$.)

I think the answer is (consistently) no.

Following Asaf's comment, let $\kappa$ be Mahlo in $L$ and let $G \subset \text{Col}(\omega,\mathord{<}\kappa)$ be an $L$-generic filter. Let $\eta$ be the least ordinal such that $\kappa < \eta$ and $L_\eta \models \mathsf{ZFC}$, and consider the model $N = L_\eta[G]$.

Then in $N$ there is an uncountable transitive model of $\mathsf{ZFC}$, namely $L_\kappa$. It suffices to show that every such model has an inaccessible cardinal.

Let $M$ be an uncountable transitive model of $\mathsf{ZFC}$ in $N$. Note that $\text{Ord}^M \ge \kappa$ by uncountability and in fact $\text{Ord}^M = \kappa$ because otherwise $L^M$ would violate the minimality of $\eta$.

The set of $L$-inaccessibles below $\kappa$ is stationary in $L$ by definition, and this stationarity is preserved by the Levy collapse forcing that adds $G$ to obtain $N$. So the model $M$ satisfies "the class of $L$-inaccessibles is definably stationary."

Therefore (because $M \models \mathsf{ZFC}$) there is an $L$-inaccessible $\alpha < \kappa$ that is a strong limit cardinal in $M$. Then $\alpha$ is inaccessible in $M$ by Jensen's covering theorem (note that $0^\sharp$ could not have been added by forcing over $L$.)