As Donu pointed out, it may happen that the general section has no zeroes.  Anyway, the  the answer to your question is *yes*. 

This is a special case of the following more general result "of Bertini type" about degeneracy loci of morphism of vector bundles, whose proof can be found in Ottaviani's book *Varietà proiettive di codimensione piccola* (in Italian, but I guess that many references in English are also available).

>**THEOREM.** Let $\phi \colon F \to E$ be a morphism between vector bundles on $X$, with $\textrm{rank} \, F=s$, $\textrm{rank} \, E=r$,  and let $D_k(\phi)$ be the set of points $x \in X$ such that $\phi_x \colon F_x \to E_x$ has rank $\leq k$.

>If $F^{*} \otimes E$ is generated by global sections, then for the general $\phi$ either $D_k(\phi)$ is empty or it has the expected codimension $(s-k)(r-k)$.

>Moreover, if $F^{*} \otimes E$ is ample and  $(s-k)(r-k) < \dim X$, then for the general $\phi$ the locus $D_k(\phi)$ is non-empty, of the expected codimension $(s-k)(r-k)$.

Now apply this result to the morphism $$\phi \colon \mathcal{O}_X \to E$$ 
induced by the section. Since $D_0(\phi)$ is exactly the locus where the section vanishes, you are done.