A derivation on an $R$-algebra $A$ is an $R$-linear map $d:A\rightarrow A$ such that, the following condition is satisfied: $$d(ab)=d(a)b+ad(b)$$ for all $a,b\in A$.
Given a manifold $M$, a derivation on $M$ is a derivation on the $\mathbb{R}$ -algebra $C^\infty(M)$, of smooth functions on $M$.
Then there is notion of derivation of vector bundle $E\rightarrow M$, which is defined to be a map of sections $\Gamma(M,E)\rightarrow \Gamma(M,E)$ satisfying certain conditions.
When we take the trivial vector bundle $M\times \mathbb{R}\rightarrow M$, the set of sections $\Gamma(M, M\times \mathbb{R})$ is precisely the smooth maps. Though $\Gamma(M, E)$ would not be an algebra.
So, the idea of taking sections looks fair. But, I don't fully understand the point. Any clarification regarding this would be useful.