Here is a quotation from page 110, section 4-1, of Chern's Lectures on Differential Geometry:
If a section $s$ of a vector bundle $E$ satisfies the condition $$ Ds = 0\ \ \ \ \ \ \ \ \ \ \ (1.36), $$ then $s$ is called a parallel section. The zero section is obviously a parallel section; yet in general nonzero parallel sections may not exist. If we express $s$ with respect to the local frame field $S$ by $s=\sum_{\alpha=1}^q \lambda^{\alpha}s_{\alpha}$, then expression (1.36) is equivalent to $$ d\lambda^{\alpha} + \sum_{\beta=1}^q \lambda^{\beta} \omega_{\beta}^{\alpha} = 0, \ \ \ \ \ \ \ \ \ \ 1\le\alpha\le q.\ \ \ \ \ (1.37) $$ This is a Pfaffian system of equations. If we let $$ \theta^{\alpha} = d\lambda^{\alpha} + \sum_{\beta=1}^q \lambda^{\beta}\omega_{\beta}^{\alpha}, $$ then $$ d\theta^{\alpha} = \sum_{\beta=1}^q \theta^{\beta}\wedge\omega_{\beta}^{\alpha} + \sum_{\beta=1}^q \lambda^{\beta}\Omega_{\beta}^{\alpha}. $$ Thus we see that if the curvature matrix of the connection $D$ is zero, then $$ d\theta^{\alpha} \equiv 0\ \ \ \ \ \mod{(\theta^1,\ldots,\theta^q)}, $$ that is, the system (1.37) is completely integrable. In this case, there exist $q$ linearly independent parallel sections. Similarly, for (1.37) to have nonzero solutions, we need to impose certain conditions on the connection.
I have several questions about this.
Let $M$ be the base manifold of the vector bundle $E$. Chern observes that (1.37) is a Pfaffian system. Naturally, I thought he meant is was a Pfaffian system on $M$. I now think he's actually thinking of it as a Pfaffian system on the product manifold $M\times \mathbb{R}^q$, where the $\mathbb{R}^q$ corresponds to the $\lambda^{\alpha}$, which are thought of as "variables". Is that right?
Chern applies the Frobenius theorem to conclude (1.37) is completely integrable. All that means to me is that there exist integral submanifolds: for each $(p,\lambda)\in M\times \mathbb{R}^q$, there is a manifold through $(p,\lambda)$ on which (1.37) vanishes. I think he then concludes from this that there are "solutions" $\lambda^{\alpha}$ of the system (1.37), and therefore one can construct parallel sections $s=\sum_{\alpha=1}^q \lambda^{\alpha}s_{\alpha}$. Is this a right? Should this be obvious? It seems like a big leap skipping lots of details. Maybe if I work though it, it'll become clearer.
He also concludes there are $q$ linearly independent parallel sections. How does he know this? My guess is that he knows there are solutions of (1.37) through any $(p,\lambda)\in M\times\mathbb{R}^q$, so there are solutions with $\lambda=(1,0,0,\ldots)$, $\lambda=(0,1,0,\ldots)$, etc., and each of these solutions must give rise to linearly independent sections, since they are identical to $s_1,s_2,\ldots$ at $p$, which are assumed to be linearly independent. Is that right?
Finally, how does he go from all of these local conclusions to global conclusions? Is it true that $q$ parallel sections must exist and be defined on the entire manifold $M$ and be linearly independent everywhere?
Yes, you're pretty much right on everything. But remember that this is all local; to get global sections may be impossible if the topology of the manifold interferes. The $s_\alpha$ are giving a local trivialization of $E$, so, yes, you're working on (a piece of) $M \times \Bbb R^q$. You're correct about choosing initial conditions for the $\lambda$'s.
What is crucial to check is that the integral submanifolds are (locally) diffeomorphic to $M$ — think of them as giving graphs of ($\Bbb R^q$-valued) functions on $M$. As you point out, any tangent vector $X$ to such an integral submanifold cannot be "vertical," because the $d\lambda^\alpha$ give a basis for the "vertical" $1$-forms (and the $\omega_\beta^\alpha$ are $1$-forms on $M$). The last comment I'll make is that you will use repeatedly the fact that linear independence (maximal rank) is an open condition, so linear independence at one point will allow you to apply the inverse function theorem to see projection from an integral submanifold to $M$ is a local diffeomorphism.