$\quad$ (Following, e.g. SBBM) Given a Lie algebra contraction $\mathfrak{g}\xrightarrow{t(\epsilon)}\mathfrak{g}_0$, one can contract a family $\{\rho_{\epsilon}:\mathfrak{g}\rightarrow gl(V_{\epsilon})\}$ of representations of $\mathfrak{g}$ (on some inner product spaces $V_{\epsilon}$) to a representation
$$ \rho_0:\mathfrak{g}_0\rightarrow gl\bigg(\underset{\longrightarrow}{\lim}V_{\epsilon}\bigg)$$
of $\mathfrak{g}_0$ on the direct limit of the $V_{\epsilon}$ by setting
$$ \rho_0(\tilde{X}_j)[v] := \lim_{\epsilon\rightarrow 0^+}[\rho_{\epsilon}(t(\epsilon)(X_j))\varphi_{\epsilon'\epsilon}(v)], $$
where $\{X_j\}$ are the generators for $\mathfrak{g}$, $\{\tilde{X}_j\}$ are the (contracted) generators of $\mathfrak{g}_0$, $\varphi_{\epsilon'\epsilon}:V_{\epsilon'}\rightarrow V_{\epsilon}$ are the transition maps from the definition of the direct limit, and where we have chosen a representative $v\in V_{\epsilon'}$ for $[v]\in \displaystyle\underset{\longrightarrow}{\lim}V_{\epsilon}=\sqcup_{\epsilon}V_{\epsilon}/\sim$ (and $\sim$ is the natural equivalence relation for taking direct limits of inner product spaces -- more details on all of these pieces can, of course, be added, if needed). This representation $\rho_0$ is called the contraction of the representations $\{\rho_{\epsilon}\}$ (which is consistent with, for example, the notion of contraction of representations given in the classic Inonu-Wigner paper on contractions).
Question: Can this procedure be extended to the universal enveloping algebras, i.e. does the contraction $t(\epsilon)$ yield a contraction $\mathfrak{U(g)}\xrightarrow{\tilde{t}(\epsilon)}\mathfrak{U(g}_0)$ (and then, of course, to representations of $\mathfrak{U(g)}$ and $\mathfrak{U(g}_0)$)?
This isn't immediately obvious to me, and I cannot seem to find any references that discuss this.
There is no difference between $U(\mathfrak{g})$-modules and $\mathfrak{g}$-modules, because they can be identified. Hence it does not matter concerning contractions. But contractions can be defined for any finite-dimensional algebra and its representations. However, the universal enveloping algebra is infinite-dimensional, which is perhaps not in the sense of Inonü-Wigner contractions. For a survey on contractions of algebras, algebraic groups etc. see here.