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.

The pure state space of the Dixmier algebra is separable, so it has many faithful representations on separable Hilbert spaces, but it certainly seems that there is no canonical representation.