If we assume $Vect_n$ means the category whose objects are $n$-dimensional real vector spaces and whose morphisms are $\mathbb{R}$-linear maps, then your question doesn't quite make sense, because the notation $\phi(f)$ requires $f$ to be invertible. If instead we assume $Vect_n$ means the core of the previous category, i.e., morphisms are isomorphisms of real vector spaces, then a positive answer to your question follows from global choice. This is essentially your reasoning involving a basis: we may define a functor on a skeleton of the source category and choose a noncanonical extension.
The usual method for expressing naturality of a representation is to describe it as the restriction of an additive endofunctor on the category of all finite dimensional real vector spaces, instead of restricting to a single dimension. The representations you get are precisely those defined by polynomial equations (i.e., algebraic representations), and the functors are direct sums of Schur functors. (Edit: A previous version of this paragraph restricted to additive endofunctors, but these are uniquely determined by what happens to the one dimensional vector space, and hence a bit too restrictive.)
In general, you can get many other representations. For example, we may choose a discontinuous automorphism $\psi$ of $GL_1(\mathbb{R})$, and take the tensor product of any nice map $GL_n \to GL_m$ with the composite of determinant with $\psi$, to get a rather awful object. Even with continuous homomorphisms, one can get things like $A \mapsto \begin{pmatrix} 1 & s \log |\det A | \\ 0 & 1 \end{pmatrix}$ for real numbers $s$, so I doubt there is a good classification.