Unfortunately, the tubular neighborhood theorem is not true in general in the holomorphic context. To see the obstruction, consider the exact sequence of holomorphic vector bundles over $E$: $0\to TE\to TS|_E \to NE \to 0$. If we were to have a holomorphic embedding of a neighborhood of the zero section of $NE$ into $S$, in particular we would obtain a splitting of this exact sequence. The obstruction to doing so is a class in $Ext^1(NE,TE)$, as can be seen by applying the left-exact functor $Hom(NE,\cdot)$ and extending on the right to an exact sequence.
To be more explicit, let $q:TS|_E\to NE$ denote the quotient map, and consider the following terms of the long exact sequence $Hom(NE,TS|_E) \to Hom(NE,NE) \to Ext^1(NE,TE)$. The first map takes $f:NE\to TS|_E$ and composes it with $q$ to get a map from $NE$ to itself, and the second is the coboundary map $\delta :Hom(NE,NE)\to Ext^1(NE,TE)$. By definition, $f$ is a splitting of the short exact sequence above if and only if $qf$ is the identity map on $NE$. By exactness, such an $f$ exists if and only if $\delta(id_N)=0$, so this class is the obstruction; it may or may not be easy to compute it in any given example. (in the last equation, "N" means "NE"... I couldn't get it to typeset properly)
Also, although I'm not an expert, it's my understanding that there's a construction known as "deformation to the normal cone", which allows one to get around the failure of the holomorphic tubular neighborhood theorem in my situations. In particular, I've heard it's often useful in intersection theory, so it may have some bearing on the problem you describe in your question.
