The standard way that SVP is formalized is such that what you ask isn't really relevant to showing $SVP\in\mathsf{NP}$.
The typical formalization of SVP is (for an arbitrary norm $\lVert\cdot\rVert$ on $\mathbb{R}^n$ --- note that the hardness of SVP can depend on the particular choice of norm):
Let $n\in\mathbb{N}$, and $\gamma\in\mathbb{R}_{\geq 0}$. An instance of SVP is a pair $(\Lambda, \gamma)$, where $\Lambda\subseteq\mathbb{R}^n$ is a lattice, and $\gamma$ a constant. We say that an SVP instance $(\Lambda,\gamma)$ is accepting if:
$$\min_{v\in\Lambda\setminus\{0\}}\lVert v\rVert \leq\gamma$$
and rejecting otherwise.
In terms of this formulation of the problem, the NP witness to a problem instance $(\Lambda, \gamma)$ is any vector $v\in\Lambda\setminus\{0\}$ such that $\lVert v\rVert \leq \gamma$.
These clearly can be efficiently described, and it should also be clear how one can efficiently verify whether $(\Lambda,\gamma)$ is accepting or rejecting, given such a witness.
Of course, your question has a broader interpretation --- can we efficiently determine whether some candidate shortest vector $v$ is "actually" the shortest vector in the lattice? I'm not an expert, but:
- I don't believe this is known in the worst case (but am not sure)
- In the average case (which is the one everyone cares about), strong enough concentration results on $\lambda_1(\Lambda)$ are known that it doesn't matter.
In particular, for most candidate "hard" distributions over lattices $\Lambda\gets \mathcal{D}$, $\lambda_1(\Lambda)$ is highly concentrated around some known value, so to "check" if some candidate vector $v$ is the shortest in some random lattice $\Lambda$, it suffices to check if $\lVert v\rVert$ is close to the known value of $\mathbb{E}_{\Lambda\gets\mathcal{D}}[\lambda_1(\Lambda)]$.
See for example Random Lattices: Theory and Practice, which includes some pointers to relevant mathematical work.
