The aim of this series of papers is to generalize the ambient approach of Duval et al. regarding the embedding of Galilean and Carrollian geometries inside gravitational waves with parallel rays. In this paper (Paper I), we propose a generalization of the embedding of torsionfree Galilean and Carrollian manifolds inside larger classes of gravitational waves. On the Galilean side, the quotient procedure of Duval et al. is extended to gravitational waves endowed with a lightlike hypersurface-orthogonal Killing vector field. This extension is shown to provide the natural geometric framework underlying the generalization by Lichnerowicz of the Eisenhart lift. On the Carrollian side, a new class of gravitational waves – dubbed Dodgson waves – is introduced and geometrically characterized. Dodgson waves are shown to admit a lightlike foliation by Carrollian manifolds and furthermore to be the largest subclass of gravitational waves satisfying this property. This extended class allows us to generalize the embedding procedure to a larger class of Carrollian manifolds that we explicitly identify. As an application of the general formalism, (Anti) de Sitter spacetime is shown to admit a lightlike foliation by codimension one (A)dS Carroll manifolds.