The Chern classes are, by definition, cohomology classes. And cocycle representatives of the Chern classes are not unique.

But it might be the case that cocycle representatives of the Chern classes are unique up to contractible choice.

The following kind-of-answer is not what I'm looking for:
One obvious way to make Chern classes of vector bundles unique up to contractible choice is to pickm once and for all, cocycle representatives of the universal Chern classes $c_n\in H^{2n}(BU)$. But that's not a good answer, because this depends on making a choice of lifting someting in $H^{2n}(BU,\mathbb Z)$ to something in $Z^{2n}(BU,\mathbb Z)$. I don't want to have to make any arbitrary choices.

So, what I'm asking is:
is there a best way to pick (up to contractible choice) a cocycle representative of the universal Chern classes $c_n\in H^{2n}(BU)$?