In Mirror Symmetry, specially in Homological Mirror Symmetry, the mirror of stable holomorphic Vector bundle is the special Lagrangian submanifold.
In symplectic geometry
A-brane = Lagrangian submanifold + flat vector bundle •
In holomorphic geometry
B-brane = complex submanifold + holomorphic bundle
There is a relation between reflexive Sheaf and Vector bundle which is used in the study of Extension theory for finding canonical metrics
A reflexive sheaf $ F$, on Kähler variety $X$ outside of codimension at least 3 is a holomorphic vector bundle,
In the study of positivity theory of direct image of Line bundles which are Vector bundles.
For example positivity of CM-bundle which is related to K-stability and Kähler-Einstein metric on Fano manifolds