Background. The Witt algebra is the Lie algebra with generators $\{l_n:n\in \mathbb{Z}\}$ with bracket given by $$[l_m,l_n]=(m-n)l_{m+n}.$$ The Virasoro algebra is the unique central extension of the Witt algebra with the bracket relation given by $$[L_n,L_m]=(m-n)L_{m+n}+\frac{m^3-m}{12}\delta_{m,-n}K.$$
In http://library.msri.org/books/Book57/files/40mason.pdf, we define Vir$^\geq=\langle L_n,K: n\geq 0 \rangle$ and Vir$^-=\langle L_n: n<0\rangle$ and construct the 1-dimensional $\text{Vir}^\geq$-module $\mathbb{C}v_{c,h}$ by defining the actions $$K.v_{c,h}=cv_{c,h}$$ and $$L_n.v_{c,h}=\delta_{n,0} hv_{c,h}$$ for $n\geq 0$ ($v_{c,h}\neq 0$ and $c,h$ are arbitrary scalars). This gives rise to the induced (Verma) module $$M_{c,h}=\mathcal{U}(\text{Vir})\otimes_{\mathcal{U}(\text{Vir}^\geq)}\mathbb{C}v_{c,h}=\mathcal{U^-}(\text{Vir})\otimes \mathbb{C}v_{c,h}.$$ Question. Is there any obstruction to performing the analogous construction for the Witt algebra where the action of the central element no longer exists? I don't see why not, since the underlying vector space for the Witt algebra decomposes as $\text{Witt}^\geq \oplus \text{Witt}^-$ (assuming these have the obvious definitions analogous to above) and the module structure still makes sense without the action of $K$.
The reason I am interested in this is because I am trying to figure out what the obstacle is in constructing a vertex algebra from the Witt algebra (Is it possible to construct a vertex algebra from the Witt algebra?), since this would be a natural example of a VA if it existed but no sources mention it; I am assuming this is due to my misunderstanding of the above construction.