How do the particles in a rigid body "know" when to accelerate?

"Rigid body" is an idealization.

Bodies that are well approximated as "rigid" have high tensile strength, high elastic modulus and high speed of sound. In this construct, "high" relates to the scale and forces of the system in question.

In reality, there is no absolutely rigid bodies. The acceleration propagates by acoustic waves with the speed of sound.

A certain body may be regarder both as rigid and as non-rigid. Imagine a church bell - you may consider it rigid when planning on how to transport it from the bell workshop to the church tower top. You have to assume it is pretty much elastic and non-rigid when you have to understand how it rings.

(There is a traditional physics practice experiment where you hit two metal cylinders together at their bases and use an electronic timer connected to them in order to determine the speed of sound in the metal. Cylinders stay connected together for the duration of the impact and the impact duration is the time the acoustic wave travels to the free ends and returns back to the impact side.)

Exactly because all portions of the body don't move instantaneously according to an applied force at a given point, that most of the noises of everyday life are produced.

The disturbance propagates as an elastic wave through the body as explained here

If the body will only vibrate for a fraction of second, but stay at rest, or will also move as a rigid body, it depends on the boundary conditions (static friction forces for example).

The cause of the vibration are atomic/molecular interactions in the solid. In a metal for example, the cloud of electrons hold the nuclei at its positions. Any attempt to increase or decrease the interatomic distance generates a restoring force as in a spring-mass system.

The vibration effect is normally only a transient that vanishes very quickly by damping, if the final effect in the rigid body is a translation, or a rotation around one of the axis of inertia.