Projective limits in vector spaces and in sets are the same so the stalk does not depend on whether you consider this as a co-presheaf of sets or vector spaces.
in both case it is just the directed projective limit of the $\mathcal{F}(U)$ for $U$ among neighbourhood of $x$, the corestriction maps $\mathcal{F}(U) \rightarrow \mathcal{F}(U')$ are all injective, so this projective limit is just an intersection: the stalk at $x$ is hence the set of functions on $X$ whose support is included in all neighbourhood of $x$... i.e. it is zero most of the time (unless the closure of $\{x\}$ is an open sets in which case it is $\mathbb{R}$).
At an informal level: sheaves can be thought of as functions and cosheaves as measures (you can integrate a sheaf against a cosheaf using a coend, multiply a cosheaf by a sheaf to get a cosheaf etc...). Functions tend to be determined by values at points (i.e. stalks) but for measure the "value at a point" is something like $\mu(\{x\})$ and it is far from being enough to understand the measure