The obvious attempt to construct a W$^*$-envelope like the C$^*$-envelope fails because if $\mathcal{M} \subseteq \mathcal{B}(\mathcal{H})$ is an injective von Neumann algebra there may not be a normal contraction from $\mathcal{B}(\mathcal{H})$ onto $\mathcal{M}$, e.g. if $\mathcal{M}$ is not purely atomic.
I think there are pretty strong obstructions to the usefulness of such an object even in the commutative case, although I can't immediately see how to use them to rule out a W$^*$-envelope altogether.
If $\mathcal{A}$ is a commutative operator algebra whose C$^*$-envelope is $\mathrm{C}([0, 1])$, then its injective envelope is the Dixmier algebra of bounded Borel functions on $[0, 1]$ modulo the ideal of functions with meagre support. See this paper of Blecher and Magajna for a proof. The Dixmier algebra is an AW$^*$-algebra that is not a W$^*$-algebra, and it has no normal linear functionals.
Since any commutative W$^*$-algebra is injective, any W$^*$-envelope of a commutative operator algebra whose C$^*$-envelope is $\mathrm{C}([0, 1])$ has to contain the Dixmier algebra and every embedding of a dual operator algebra in its W$^*$-envelope would have to factor through the Dixmier algebra.