Yes. The easiest case is when $X$ is affine, say $X=\mathrm{Spec}(A)$. Then $E$ is associated to some locally free $A$-module $M$, and can be realized as $E = \mathrm{Spec} (Sym_A(M^\vee))$.

More generally, if $X$ is quasiprojective, take an ample line bundle $L$ on $X$. Then $P(E) = P(E\otimes L)$, and replacing $E$ by a sufficiently high twist by $L$, the line bundle $O_{P(E)}(1)$ is ample.

Note that you can realize $E$ as an open subset of the projective bundle $P(E\oplus 1)$, so (quasi)projectivity of the latter implies that of the former.