Yes, this is true.
The space of crossed homomorphisms from $G$ to $\Gamma$ could be identified, by taking graphs, with the space of subgroups of $\Gamma\rtimes G$ which intersect $\Gamma$ trivially and project onto $G$. You ask for discreteness of this space modulo the $\Gamma$-conjugation action. Every such subgroup is the image of an homomoprhism $G\to \Gamma\rtimes G$, thus may be identified as an element of the space $\text{Hom}(G,\Gamma\rtimes G)$ modulo the action of $\text{Aut}(G)$ by pre-composition. Let me write $X$ for the subspace of $\text{Hom}(G,\Gamma\rtimes G)$ consisting of homomorphisms with full image after projecting to $G$. It is enough to show that $X$ is discrete mod the action of $\text{Aut}(G)\times \Gamma$, where the first factor acts by pre-composition and the second by post-composition via its inner action. In fact, it is already discrete mod $G\times \Gamma$, where $G$ acts by precomposition via its inner action. Indeed, by the defining property of $X$, the $G\times \Gamma$-orbits are the same as the $\Gamma\rtimes G$-orbits for the postcomposition inner action, so this follows from the discreteness of $\text{Hom}(G,\Gamma\rtimes G)$ mod $G\rtimes \Gamma$.