Let $M$ be an arbitrary smooth manifold of dimension $n$. For simplicity, let's assume that $M$ is boundary-less. Can we construct a gaussian random field on $M$?
If the result is not true for arbitrary $M$, is there a theorem that provides minimal assumptions on $M$ for which we always can find a gaussian random field?
Yes. Robert Adler has written some books on Gaussian random fields that I enjoyed perusing. For specific examples of Gaussian random fields that are guaranteed to exist on closed manifolds under minimal assumptions you could look at http://arxiv.org/abs/1207.6419 and the references therein.