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, or the empty category. (I'm a little confused as to whether the empty category should be regarded as being idempotent complete.) 

Suppose $C$ is neither of these, 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).