The symplectic group acts (n)-transitively on connected manifolds of dim $\geq 2$. See Michor and Vizman
n-Transitivity of certain diffeomorphism groups
https://arxiv.org/pdf/dg-ga/9406005.pdf
(1)-transitivity is according to them due to Hatakeyama
Some notes on the groups of automorphisms of contact and symplectic structures