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 Towards a Lie theory for locally convex groups.

Proposition 3.17 in Slice theorem and orbit type stratification in infinite dimensions 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.

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.

Proposition 3.17 in Diez and Rudolph - Slice theorem and orbit type stratification in infinite dimensions 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.