No. Let $C$ be any category with a zero object. If $F : C \to \text{Set}$ is a cosheaf, then in particular it sends the zero object to the empty set. But since the zero object is also the terminal object, every object in $c$ is equipped with a morphism $c \to 0$, from which it follows that $F(c)$ is equipped with a morphism to the empty set. Hence $F(c)$ is itself empty, so $F$ is the trivial cosheaf. Now take $C$ to be, say, the category of finite-dimensional vector spaces.
Edit: In fact the free example suffices. Let $C$ be the free category on a pair of parallel morphisms $f, g : c \to d$. Then the coproduct $d \sqcup d$ exists and both inclusions $d \to d \sqcup d$ are isomorphisms. It follows that if $F$ is a cosheaf then $F(d)$ is the empty set, from which it follows that $F(c)$ must also be the empty set and $F(f) = F(g)$ must be the unique morphism $\emptyset \to \emptyset$, so $F$ is again the trivial cosheaf.