There's a well-known result (for example, Th. 14.2.7 in tom Dieck's book) that the category of principal $ \operatorname{GL}_n(\mathbb{R}) $-bundles and bundle maps is equivalent to the category of $ n $-plane bundles and bundle maps by sending a principal $ \operatorname{GL}_n(\mathbb{R}) $-bundle to to its associated fiber bundle.
A number of people have mentioned to me that a similar result holds between the categories of $ n $-plane bundles over paracompact spaces, and principal $ O(n) $-bundles over paracompact spaces. However, I can't seem to find a reference for this. I'm wondering if anyone knows of a reference for this (of if it's actually not true).
First, notice that the claimed equivalence between principal $\operatorname{GL}_n(\mathbb{R})$-bundles and $n$-plane bundles holds only if you consider only those vector bundle maps which are invertible, i.e. they are linear isomorphisms on fibers.
Your claim about principal $O(n)$-bundles is not true. The category of principal $O(n)$-bundles is equivalent to the category of $n$-plane bundles endowed with an Euclidean metric, and having as morphisms bundle maps which are linear isometries fiberwise.
What is true is that on a $n$-plane bundle on a paracompact space you can always put such an Euclidean metric using a partition of unit, but then an arbitrary vector bundle map is not necessarily an isometry, and in this case it does not induce a map of principal $O(n)$-bundles.