It is true that in solids the atoms and molecules of the lattice are held together in a quantum mechanical state which depends on electromagnetic fields and exchanges. Certainly this sets a limit to how fast a signal can travel in a solid. In metals, where electrons are shared by the whole lattice, the signal that an electron has been removed travels with the velocity of light through the metal.
But in a lattice with stable configurations of the atoms, connected with electromagnetic springs,an atom can only vibrate in its site location about a center, and transfer the energy vibrationally.
This has been studied using phonons
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, like solids and some liquids. Often designated a quasiparticle,1 it represents an excited state in the quantum mechanical quantization of the modes of vibrations of elastic structures of interacting particles.
Also this link is readable on phonons.
A phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because phonons play an important role in many of the physical properties of solids, such as the thermal conductivity and the electrical conductivity. In particular, the properties of long-wavelength phonons gives rise to sound in solids -- hence the name phonon. In insulating solids, phonons are also the primary mechanism by which heat conduction takes place.
As a rule of thumb , the impulse/kick on a solid object will travel through its lattice at the speed of sound in that solid .