Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the topology of uniform convergence on compact subsets of $X$).
Question. Is the space $X_c'$ sequential (equivalently, a $k$-space)?
Yes. In the next paragraph I will show that if $X$ is a Fréchet space (without requiring separability) then $X'_c$ with the compact-open topology is a $k$-space. As you note, this implies sequentiality if $X$ is separable.
The compact-open topology on $X'_c$ is the same as the finest topology agreeing with $\sigma(X'_c, X)$ on equicontinuous sets by the "Banach-Dieudonné" theorem (see Schaefer's Topological Vector Spaces, Chapter IV, Theorem 6.3). As Fréchet spaces are barrelled, all $\sigma(X'_c,X)$-bounded sets, and therefore all $\sigma(X'_c,X)$-compact sets, are equicontinuous. We also have, by the Bourbaki-Alaoglu theorem, that closed equicontinuous sets are $\sigma(X'_c,X)$-compact. Therefore the compact-open topology is the finest topology agreeing with $\sigma(X'_c,X)$ on compact sets, i.e. the Kelleyfication of the $\sigma(X'_c,X)$ topology, so $X'_c$ is a $k$-space.
