Dear Benjamin, the statement that holomorphic sections are dense in the smooth sections is  false, already for the trivial bundle of rank one $E=X\times \mathbb C$ over $X=\mathbb C$. Indeed on any non-empty set $U \subset \mathbb C$ it is impossible to approximate the $C^{\infty}$ function $\bar z$ by holomorphic functions since the limit of a sequence of holomorphic functions on $U$ is a holomorphic function on $U$. [The limit is to be understood in the sense of uniform convergence on compact subsets of $U$. There is also an $L^2$- version stating that $L^2(U)\cap \mathcal O(U)$ is a Hilbert subspace  of  $L^2(U)$, so that a sequence of holomorphic functions converging only in the $L^2$ sense nevertheless has a holomorphic limit. ]