Definition. Let $ \pi :E \to M $ be a smooth vector bundle of rank $r$. A subbundle of rank $k$ is a disjoint union $E'$ of $k$-subspaces, one for each fiber $E_p$ , such that $E'$ is an embedded submanifold of $E$, and the couple $(E',\pi_{|E'})$ is a vector bundle of rank $k$ over $M$.
I want to prove the following
Proposition. Let $E'$ be a disjoint union of $k$-subspaces $E_p '$, one for each fiber $E_p$. Then, $E'$ is a subbundle of $(E,\pi)$ if and only if for every trivializing atlas $\{(U_\alpha , \varphi_\alpha)\}$ of $M$ with respect to the vector bundle $(E,\pi)$, we can choose local trivializations $\chi_\alpha : \pi^{-1} (U_\alpha) \to U_\alpha \times \mathbb R ^r $ such that, for every $\alpha$ and for every $p\in U_\alpha$, $\chi_\alpha(E'_p)=\{p\} \times (\mathbb R ^k \times {0})$.
Direction $\Leftarrow $ is almost obvious. What about direction $\Rightarrow$? Is it true?
No, it's not true. For example, the tangent bundle $T\mathbb S^2$ is a subbundle of the trivial bundle $\mathbb S^2\times\mathbb R^3=T\mathbb R^3|_{\mathbb S^2}$. The latter has a trivializing atlas consisting of one element, namely $(\mathbb S^2, \text{id})$. However, because $T\mathbb S^2$ is not trivial, it's not possible to find a local trivialization of the trivial bundle over all of $\mathbb S^2$ that also trivializes $T\mathbb S^2$.
What is true is the following weaker result:
Proposition. Let $E'$ be a disjoint union of $k$-subspaces $E_p '$, one for each fiber $E_p$. Then, $E'$ is a subbundle of $(E,\pi)$ if and only if for every $p\in M$, there exists a neighborhood $U$ of $p$ and a local trivialization $\chi\colon \pi^{-1}(U)\to U\times \mathbb R^r $ such that, for every $p\in U$, $\chi(E'_p)=\{p\} \times (\mathbb R ^k \times {0})$.
To prove the $\Rightarrow$ direction, start with a local frame $(E_1,\dots,E_k)$ for $E'$ in a neighborhood of $p$, and then extend it to a local frame $(E_1,\dots,E_r)$ for $E$ on a (possibly smaller) neighborhood of $p$. The map $\psi\colon U\times\mathbb R^r\to \pi^{-1}(U)$ defined by $\psi(p,v) = v^1E_1|_p + \dots v^r E_r|_p$ is easily shown to be a diffeomorphism, and its inverse is the desired local trivialization.