Assume you have a vector bundle $\Pi: E \rightarrow M$ where $E$ is the total space, $M$ is a compact manifold. Assume you know it is parallelizable. Let $\psi_i : \Pi^{-1}U_i \rightarrow U_i \times \mathbb{R}^n$ be local trivializations for some covering $\{U_i\}_{i=1}^m$. I am trying to build a non-vanishing section section $\alpha$ so that
$$|\alpha|_{C^1} \leq \max_i||\psi_i||_{C^1}$$
where $C^1$ denotes the $C^1$ supnorm (there are many ways to define this, one is just to take the maximum of supnorm of derivatives of the object in hand with respect to a chosen coordinate system).
One way that came to my mind is to try to pull back constant sections on $U_i \times \mathbb{R}^n$. What I mean is try to build non-vanishing section $\alpha$ so that
$$\alpha|_{U_i} = (\psi_i)^*\eta_i$$
where $\eta_i$ is just a constant section on $U_i \times \mathbb{R}^n$ so that $|\eta_i|_{\infty} \leq 1$. Then the estimate is satisfied. This would be a section all right but I am not sure how to force it to be non-vanishing.
Also for clarity, I should probably tell the real reason why I need this. Assume the setting above but $E$ is a parallelizable on a sequence of open sets $V^k \subset M$. For each $k$, I want to build a basis of sections $\{\alpha^k_i\}$ for $E$ so that $|\alpha^k_i|_{C^1} \leq D$ where $D$ is a constant that does not depend on $k$. I would actually also like to have so that that their norm is bounded below uniformly as well in the sense that $|\alpha^k_i(p)|\geq C >0$. I also state this specifically since it seems problematic in the case when you try to build such a section for a sequence of disks inside $S^2$ which converges to $S^2-$northpole.