This can be done if you extend the triangulation of your 3-manifold $M$ to a triangulation of a 4-manifold $W$ whose boundary is $M$. You can find the basic idea in Chapter 9 of [these draft notes][1] on my web page, and there are very explicit details for the associated Hilbert space in [this arXiv preprint][2] (with Z. Wang). (In the latter paper we use cubes instead of simplices to make computer implementation easier.) There's also a summary in the [notes from a talk][3].
The main idea is that the Witten-Reshetikhin-Turaev TQFT can be reinterpreted as a 3+1-dimensional TQFT, and the 3+1-dimensional version is "fully extended" (goes all the way down to points). Once you have a fully extended TQFT, standard techniques allow you to construct a state sum model. In this case, the state sum you get is a modified version of the Crane-Yetter state sum. (You would get a Turaev-Viro state sum for a fully extended 2+1 dimensional TQFT.)
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EDIT:
If you don't want to wade through the above sources, here's the explicit formula.
Let $C$ be a modular ribbon category.
Let $D= \sqrt{\sum_i d_i^2}$, the square root of the sum of the squares of the quantum dimensions.
Choose a handle decomposition of $W^4$.
Keep in mind is the case where the $i$-handles are thickenings of the $i$-cells
of some cell decomposition of $W$, such as the dual cell decomposition to a triangulation.
Let ${\cal H}_i$ be the set of $i$-handles.
Let ${\cal L}_2$ be the set of labelings of the 2-handles by simple objects of $C$.
For fixed $\alpha\in{\cal L}_2$, let ${\cal L}_1(\alpha)$ denote the set of labelings of the 1-handles by orthogonal
basis elements of the associated vertex spaces.
(To each 1-handle is associated a $C$-picture on the linking 2-sphere,
that is, a collection $c$ of
ribbon endpoints on the 2-sphere labeled by simple objects
$a_1\otimes \cdots\otimes a_m$ according to $\alpha$.
The "vertex space" is $\hom_C(1, a_1\otimes\cdots\otimes a_m)$ or,
more canonically, the vector space associated to $(B^3; c)$, where we think
of $B^3$ as the normal fiber to the core of the 1-handle.)
Then
$$Z(W^4) = \sum_{\alpha\in{\cal L}_2} \sum_{\beta\in{\cal L}_1(\alpha)}
\prod_{h_4\in {\cal H}_4} D
\prod_{h_3\in {\cal H}_3} D^{-1}
\prod_{h_2\in {\cal H}_2} D^{-1} \,\text{Loop}(h_2, \alpha)\quad\quad\quad\quad$$
$$\quad\quad\quad\quad\prod_{h_1\in {\cal H}_1} D \,\text{Th}(h_1, \beta)^{-1}
\prod_{h_0\in {\cal H}_0} D^{-1} \, \text{Link}(h_0, \alpha, \beta) , $$
where
* $\text{Loop}(h_2, \alpha)$ is the loop value (quantum dimension) of the simple object $\alpha(h_2)$;
* $\text{Th}(h_1, \beta)$ is the evaluation of generalized theta graph with the two vertices
labeled by $\beta(h_1)$ and $\overline{\beta(h_1)}$ (in the case of a generic cell decomposition where each 1-cell is incident to three 2-cells, this is just an ordinary theta graph); and
* $\text{Link}(h_0, \alpha, \beta)$ is the evaluation of the graph in $S^3$ corresponding to the link
of the 0-handle $h_0$, labeled according to $\alpha$ and $\beta$.
If $W$ has a boundary $M$ then we can place a labeled link $L$ in the boundary, and the labels of the link play a role similar to the labels of the 2-handles above. In this case the state sum computes $WRT(M, L)$.
For a generic cell decomposition of $W$ (i.e. dual to a triangulation), the $\text{Link}(h_0, \alpha, \beta)$ factor is an evaluation of a labeled ribbon graph which looks like the 1-skeleton of a 4-simplex. If we resolve the five 4-valent vertices oin this graph into pairs of 3-valent vertices, then this becomes the "15-j symbol" used by Crane and Yetter. (Keep in mind that Crane and Yetter use a different normalization which obscures the relation to the WRT invariant.)
On the other hand, if we choose a handle decomposition of $W$ with a single 0-handle and several 2-handles (no 1- 3- or 4-handles), then it is easy to see that the above state sum reduces to the Reshetikhin-Turaev surgery formula. The factor $\text{Link}(h_0, \alpha, \beta)$ is the generalized Jones polynomial (evaluated at a root of unity) in this case.
A similar variant yields the Turaev "shadow" state sum.
[1]: http://canyon23.net/math/tc.pdf
[2]: http://front.math.ucdavis.edu/1104.2632
[3]: http://canyon23.net/math/talks/ucb%20frg%20200901.pdf