The following is basically thoroughly usless general nonsense. It's main problem is that it lacks geometry. Even so,  it kind of does the job and it is probably worth mentioning. 

  Let $G$ be the structure group of your vector bundle and let $G^\delta$ be $G$ with the discrete topology.  I will assume that $G$ is connected (for example, $G$ could be $U(n)$ or $SO(n)$).

By work of Tudor Ganea, there are maps of connected spaces
$$
BG^\delta = B_0 \to B_1 \to B_2 \to \dots
$$
and compatible maps $B_i \to BG$. Here $BG$ denotes the classifying space of $G$.  Furthermore, the map $BG_k \to BG$ has homotopy fiber identified with
$$
\Sigma^k (G/G^\delta) \wedge G\wedge \cdots \wedge G
$$
where there are $k$-factors of $G$ in the above smash product and $G/G^\delta$ is shorthand for  $\text{fiber}(BG^\delta \to BG)$ (where "fiber" means homotopy fiber. 

In particular, the fact that $G$ is connected implies that the map $B_k \to BG$ is  $(2k+1)$-connected. So the limit map $\text{lim}_k B_k \to BG$ is a homotopy equivalence. 

Thus if $X\to BG$ is the classifying map of my vector bundle, I get in principle an obstruction theory for deciding when my bundle admits a flat reduction. The obstructions are encoded in a spectral sequence associated with this tower.

The classifying map of my vector bundle is given by a map $X\to BG$. It has flat reduction if and only if it factors up to homotopy through $BG^\delta = B_0$.  As an intermediate step, we can ask for which $k$ does it factor through $B_k$. Such a lift $X\to B_k$ I propose to call this a *$k$-flat* structure on the given vector bundle.

Incidentally, more general nonsense shows that $B_k = BG_k$ for a suitable topological group $G$ and the above sequence of spaces is given by applying the classifying space functor to a sequence of homomorphisms
$$
G^\delta := G_0 \to G_1 \to G_2 \to \cdots
$$
with $\lim_k G_k \to G$ an equivalence. 
In particular, a $k$-flat structure amounts to nothing more than a reduction of structure group to $G_k$.