The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is wellknown to be a modular tensor category. Can these modular tensor categories also be obtained as representation categories of vertex operator algebras?
This answer is related to my answer here: Duality between orbifold and quasiHopf algebra (twisted quantum doubles) and the comment by Scott.
Every finite group $G$ can be embedded in some symmetric group $S_n$ and then you can take for example the $n$fold product of the $E_8$ lattice VOA, for which the representation category is trivial. Then I guess the $G$ orbifold of this VOA should have $\mathrm{Rep}(D^\omega(G))$ as a representation category. This would be true in the setting of conformal net, see my answer here: Duality between orbifold and quasiHopf algebra (twisted quantum doubles) but I am not sure how much a result like: "an extension of a rational VOA is given by a special symmetric Frobenius algebra $A$ in the category of Representations and the Representation category of the extension is given by the the subcategory of dyslexic (local) module category of $A$." is established in VOA.
In this case the results mentioned in http://arxiv.org/abs/0909.2537v1 also show that every VOA with $\mathrm{Rep}(D^\omega(G))$ as representation category is a $G$ orbifold of a holomorphic VOA.
Update I could still not find an exact statement in VOA, but in Alexei Davydov wrote in http://arxiv.org/abs/1312.7466 (p.2.)
Our examples come from permutation orbifolds of holomorphic conformal field theories (CFTs whose state space is an irreducible module over the chiral algebras) . It is argued in [30] (see also [35]) that the modular category of the $G$orbifold of a holomorphic conformal field theory is the so called Drinfeld (or monoidal ) centre $\mathcal Z(G,\alpha)$, where $\alpha$ is a 3cocycle of the group $G$. It is also known that the cocycle $\alpha$ is trivial for permutation orbifolds (orbifolds where the group $G$ is a subgroup of the symmetric group permuting copies in a tensor power of a holomorphic theory). he assumption crucial for the arguments of [30] is the e xistence of twisted sectors. This assumption is known to be true for permutation orbifolds [1].
This statement (if true, I did not yet check [1] (http://link.springer.com/article/10.1007/s002200200633)) gives exactly the answer the ops question, like I guessed above, namely:
Let $G\subset S^n$ and and let $(V^{\otimes n})^G$ the $G$permutation orbifold of $V^n$ where $V$ is a holomorphic VOA, then $\mathrm{Rep}((V^{\otimes n})^G) \cong \mathrm{Rep}(D(G))$.
Update: The statement "an extension of a rational VOA is given by a commutative algebra $A$ (Thm 3.2) in the category of Representations and the Representation category of the extension is given by the the subcategory of dyslexic (local) module category of $A$. (Thm 3.4)" appeared now in http://arxiv.org/abs/1406.3420

$\begingroup$ The following link for reference [1] (BarronDongMason) has no paywall: arxiv.org/abs/math/9803118 $\endgroup$ – S. Carnahan♦ Mar 10 '14 at 9:43

$\begingroup$ Okay, I've had a look at the references, and I have not seen a proof that the cocycle is trivial. It is an assumption in Kirillov's paper [30] (arxiv.org/abs/math/0104242), and is simply claimed to be "known" in the paragraph of Davydov's paper that you quoted. The true fact in BarronDongMason is that the $\sigma$twisted module category is semisimple with one irreducible object for any permutation $\sigma$ (and they give an explicit construction of the irreducible twisted module). $\endgroup$ – S. Carnahan♦ Mar 10 '14 at 10:43

$\begingroup$ I could neither find aproof about my last statement in the literature. But I would still bet it is true! $\endgroup$ – Marcel Bischoff Mar 10 '14 at 14:33

$\begingroup$ In Section 4.2. Davydov also refers to [30] for a proof. $\endgroup$ – Marcel Bischoff Mar 11 '14 at 9:33

$\begingroup$ I agree that the last statement is likely to be true, but I am fairly certain that it is still open for $n \geq 3$, and that Davydov's claims are a bit too optimistic. $\endgroup$ – S. Carnahan♦ Mar 11 '14 at 12:11
A lot has happened in the last four years, and we now have lots of positive results.
The current state of knowledge is given in EvansGannon, "Reconstruction and Local Extensions for Twisted Group Doubles, and Permutation Orbifolds". In particular, if $G$ is a finite solvable group, then $D(G)$ (and more generally, any twist $D^\omega(G)$) is the representation category of some vertex operator algebra (in particular, the fixed points of a $G$action on some holomorphic vertex operator algebra).
For nonsolvable groups, the result you want would follow from the conjectured regularity of fixed points (i.e., a suitable generalization of CMiyamoto).
Oddly enough, it turns out that permutation orbifolds can have nontrivial twists. This is discussed in the EvansGannon paper, and earlier in JohnsonFreyd's "The Moonshine Anomaly".