As Mark Grant pointed out, there is no such example when $E$ is the tangent bundle of a smooth four-dimensional manifold because orientable smooth four-manifolds are spin${}^c$, so $W_3 =0$ and therefore $w_3 = 0$.

The Wu manifold $X = SU(3)/SO(3)$ is a compact, smooth, five-dimensional manifold with total Stiefel-Whitney class $w(X) = 1 + w_2(X) + w_3(X)$ (in particular, it is an example of a non-spin${}^c$ manifold). In fact, $H^*(X; \mathbb{Z}_2) \cong \bigwedge(w_2(X), w_3(X))$.

<!--On a smooth compact orientable five-dimensional manifold, the only potentially non-trivial Stiefel-Whitney number is $w_2w_3$. As there are no Pontryagin numbers in dimension five, we see that there is an injective map $\Omega^{SO}_5 \to \mathbb{Z}_2$ given by the Stiefel-Whitney number $w_2w_3$. As $w_2(X)w_3(X) \neq 0$, we see that the map $\Omega^{SO}_5 \to \mathbb{Z}_2$ is an isomorphism, and therefore $\Omega^{SO}_5$ is generated by $[X]$.-->