You shouldn't expect vector bundles to form an abelian category because they are projective module objects over local ring objects in a category of sheaves over a space. In other words, there is really no more reason to expect this of vector bundles than you would of a category of finitely generated projective modules over a ring.

You shouldn't expect vector bundles to form an abelian category because they are projective module objects over local ring objects in a category of sheaves over a space. In other words, there is really no more reason to expect this of vector bundles than you would of a category of finitely generated projective modules over a ring.

It might help to add that taking stalks at a point is an exact functor (it preserves all colimits and all finite limits), so kernels and cokernels would be preserved under taking stalks at points. Moreover, if the space is sober, then taking stalks collectively reflects kernels and cokernels as well. This should be a reassurance as to whether one is computing kernels and cokernels correctly.