By definition, a distribution on $U\subset R $ is a continuous linear functional on $C_{c}^{\infty}(U)$ where $C_{c}^{\infty}(U)$ is space of test function.
So in order to define a distribution we have to define a topology in $C_{c}^{\infty}(U)$.
Now let $\Gamma(E)$ be a space of section of a vector bundle $E\rightarrow M$.
How we define continuos linear functional in this space of section?