Is there any closed form expression for Rényi entropy of a set variables with multivariate Gaussian distribution?
Yes. First, do a change of variable in the integral to convert it to the Renyi entropy of a set of uncorrelated Gaussians with standard deviation $1$. The integral now splits into a product of 1-dimensional integrals, where each one is the Renyi entropy of a 1-dimensional Gaussian.
An alternative approach is to write the integral in polar co-ordinates and split the integral into the product of a spherical integral (which is equal to something like the determinant of the covariance matrix multiplied by the volume of a sphere) and a radial integral. The radial integral can be written in terms of gamma functions (or, beta functions) using the right change of variable.