We can replace $C$ with its idempotent completion WLOG, so the question becomes: for $C$ an idempotent complete $k$-linear category, when is every ($k$-linear) presheaf $C^{op} \to \text{Vect}$ representable? 

The answer is: iff $C$ is the zero category, by which I mean the $k$-linear category with only the zero object. (Note that this is the idempotent completion of the empty category, according to the second definition but not the first, which shows that they are not equivalent!) 

Suppose $C$ is not the zero category, so it has at least one nonzero object $c$. Then the representable presheaf $\text{Hom}(-, c) : C^{op} \to \text{Vect}$ takes at least one nonzero value. Now consider the presheaf $F(-) = \text{Hom}(-, c) \otimes W$ for an infinite-dimensional vector space $W$. We want to show that $F$ is not representable. It will suffice to show that $\text{Hom}(F, -)$ does not preserve colimits. 

$F$ itself is the filtered colimit of the presheaves $\text{Hom}(-, c) \otimes V$ as $V$ runs over all finite-dimensional subspaces of $W$, and if $\text{Hom}(F, -)$ preserved this filtered colimit then every natural transformation $F \to F$ would factor through $\text{Hom}(-, c) \otimes V$ for some finite-dimensional $V$. But the identity natural transformation does not factor in this way, as we can see by plugging in $(-) = c$ (since $c$ is nonzero, $\text{End}(c)$ is also nonzero).