Schubert2 has all of the basic operations on vector bundles and their Chern classes built in. A full list of all of the available operations can be found in the documentation for AbstractSheaf. A few examples:
Direct Sums:
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Note that the Chern class of $S+Q$ is the same as that of the trivial bundle, since $S$ and $Q$ fit into an exact sequence whose middle term is trivial (see Prop 5.5).
Tensor Products:
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Duals:
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Note that B3 is the tangent bundle to $\mathbb{G}(2,4)$.
Pullbacks:
Currently Schubert2 has few morphisms implemented, but given a morphism of abstract varieties, it is easy to pull back vector bundles:
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