Let the matrix $V$ be Vandermonde of size $n \times n$:
$$V(x_1, \dotsc, x_n) = \begin{bmatrix} 1 & x_1 & x_1^2 & \dotsb & x_1^{n-1} \\ 1 & x_2 & \ddots & & \vdots\\ \vdots \\ 1 & x_n & x_n^2 & \dotsb & x_n^{n-1} \end{bmatrix}$$
for $x_1, \dotsc, x_n \in \mathbb{C}$. I have searched unsuccessfully for results about $\|V(x_1, \dotsc, x_n)\|_2$.
The only thing I can think of is that $\|V\|_\infty$ is the largest absolute row sum. That is $1 + |x_*| + \dotsb + |x_*|^{n-1}$, where $x_*$ is the $x_i$ with largest modulus. We could also write it as $\displaystyle \frac{|x_*|^n - 1}{|x_*| - 1}$. The infinity norm bounds the two norm:
$$\frac{1}{\sqrt{n}} \|V\|_\infty \leq \|V\|_2 \leq \sqrt{n} \|V\|_\infty$$ $$\frac{1}{\sqrt{n}} \frac{|x_*|^n - 1}{|x_*| - 1} \leq \|V\|_2 \leq \sqrt{n} \frac{|x_*|^n - 1}{|x_*| - 1}$$
Can we do better?