I'm not aware of a precise characterization of when the orbits are initial submanifolds in infinite dimensions. In fact, the manifold structure on the orbits is a hard problem even for $G$-actions on a coset space $G / H$, see Problem IX.3 in [Neeb - Towards a Lie theory for locally convex groups](https://arxiv.org/abs/1501.06269).

Proposition 3.17 in [Diez and Rudolph - Slice theorem and orbit type stratification in infinite dimensions](https://arxiv.org/abs/1812.04698) shows that the orbits of Fréchet actions are embedded submanifolds if the action is proper, the stabilizer is a Lie subgroup and the linearization of the action has a tame inverse family. However, this proposition does not apply in your situation as the action is not proper in general (for example, if $k=0$, the zero distribution has the whole diffeomorphism group as stabilizer). Maybe, it works if you restrict attention to a subset of sufficiently rigid distributions.