A (rank $n$) vector bundle over a space $X$ is the "same thing as" a principal $Gl_n$ bundle over $X$. Now, consider the functor which assigns to each space $X$ the (discrete) group $Hom\left(X,Gl_n\right).$ We may regard this in fact to be a functor into groupoids, which happens to land in groups. This functor is not a stack, but we can take its stack completion in a universal way (i.e. stackify it) and the resulting functor $BGl_n$ assigns a space $X$ the groupoid of principal $GL_n$-bundles over $X$. Concretely this functor is calculated by taking the (weak) colimit over all (open) covers $U$ of $X$ of $Hom\left(X_U,Gl_n\right),$ where the latter is the groupoid, whose objects are continuous homomorphisms from $X_U$- the associated Cech groupoid- to $Gl_n$ (regarded as a topological groupoid with one object), and whose arrows are given by continuous natural transformations. It's straight forward to verify that for each cover $U$ this is simply a collection of continuous maps $$g_{ij}:U_i \cap U_j \to GL_n,$$ satisfying a cocycle condition, i.e. cocycle data for a principal $Gl_n$-bundle over the cover $U$. This is how to see that the resulting stack actually assigns the groupoid of all principal $Gl_n$-bundles. Now all of this was done for stacks with respect to the "open cover Grothendieck topology". But, there is another Grothendieck topology on compactly generated Hausdorff spaces, which is in may ways more natural. I call it the "compactly generated Grothendieck topology" here: http://arxiv.org/abs/0907.3925 . It is a Grothendieck topology such that every open cover is still a cover, but in general, there are MORE covers. Basically, a non-open covering of a space becomes a cover for this Grothendieck topology if and only if its restriction to each compact subset can be refined by an open covering. If we instead take the stack completion of $$X \mapsto Hom\left(X,Gl_n\right)$$ with respect to the compactly generated Grothendieck topology, what one gets is the concept of principal $Gl_n$-bundles which are only locally trivial when restricted to compact subsets, and using the corresponding cocycles, one can build a "weak vector bundle" in your sense. In particular, this collapses to the definition of an ordinary vector bundle when the base is locally compact Hausdorff.
In fact, such a "weak" principal $Gl_n$ bundle is the same as a continuous homomorphism $$X_V \to Gl_n$$ where $X_V$ is the Cech groupoid associated to such a generalized cover (the concept of a Cech groupoid makes sense for non-open covers). This induces a continuous map between (fat) geometric realizations $$||X_V|| \to BGl_n,$$ and I prove in the same paper I referenced above that $$||X_V||$$ is always weakly homotopy equivalent to $X$. So this should allow you to do characteristic classes as well.
