Let $T$ be a continuous functor between two different categories of finite dimensional vector spaces, $\mathcal{C}$ and $\mathcal{D}$.
Then we would like to define a functor between categories of vector bundles whose typical fibers lie in $\mathcal{C}$ or respectively in $\mathcal{D}$.
In Atiyah's K-Theory 1964 book, he defines the functor $T$ acting on a vector bundle $p:E\to X$, as a set, to be the union of the fibers $$ T(E):=\bigcup_{x\in X} T(p^{-1}(\{x\}) $$
and then goes on to define a topology on $T(E)$ which indeed makes it a vector bundle.
In other sources (e.g. Milnor & Stasheff), $T(E)$ is defined rather as the disjoint union: $$ T(E):=\coprod_{x\in X} T(p^{-1}(\{x\}) $$ though they then follow a similar construction for the topology of $T(E)$.
My question is: Is there a way to use the natural topology that comes from the disjoint union construction, and thus only define $T(E)$ as the above disjoint union with its natural topology?
For me the topology of each $T(p^{-1}(\{x\})$ is not so clear, because $T$ is a functor of vector categories and not of $Top$, so any topology that $p^{-1}(\{x\}$ has as a subspace of $E$ is "lost" when applying $T$ on it.
If the answer is "no", is there a somewhat more natural or "universal" way to define $T$ acting on the category of vector bundles?