I would like to know what the exact definition of a $\mathbb{Z}$-graded (smooth) vector bundle is. There is an obvious answer to this if only finitely many components of a graded vector bundle are nontrivial.
In general I suppose that a graded vector bundle over a smooth manifold $M$ is a graded object in the category of vector bundles over $M$ which you can equivalently think of as just a collection of vector bundles $(E_i)_{i\in \mathbb{Z}}$ over $M$. Maybe you should additionally require that for every $p\in M$ you can find an open neighborhood $U$ such that all ${E_i}_{|U}, i\in \mathbb{Z}$ are trivial.
I´ve also seen authors using the notation $E=\bigoplus_{i\in \mathbb{Z}}E_i$, so maybe you rather want to view a graded vector bundle as a bundle of graded vector spaces with fibers $\bigoplus_{i\in \mathbb{Z}}{E_i}_p,\:p\in M$?
Many thanks for your help!