The sheaf of smooth functions on a manifold is fine and hence soft, so we can extend sections on closed subsets to global sections. However, it is not generally flabby (flasque): local sections on open subsets do not in general extend to global sections.
Modules over the sheaf of smooth functions are also soft, but not in general flabby. Since the sheaf of vector fields is such a module, we wouldn't usually expect vector fields on open subsets to extend to global sections.
For example, if you have any sort of open (proper) subset which is diffeomorphic to an open ball, just take a smooth functions that blows up as you go to the boundary of the ball. Then you can rescale any vector field that does not decay to zero at the boundary to a vector field which fails to globalize.
As mentioned by Ben McKay in the comments, you can have a look at Charles Fefferman's paper proving a sharp version of the Whitney extension theorem. It's available here.