Is there a delay between force and acceleration?

In the real world, you push the atoms in the back end of the object, which push the atoms in the next layer, etc. This means that the front end of the object won't start accelerating until $t = t_0 + \Delta x/c_s$, where $\Delta x$ is the length of the object and $c_s$ is the speed of sound in the material of the object.

If you consider the object to be one atom only, you end up having to define when your push starts, since the force you apply ultimately is an electromagnetic force, which has an infinite extend, and thus applies even from far away. When you define your push to start at $t_0$, your acceleration by definition starts here as well.

Because I like drawing, here's a drawing of you pushing an object from far away. For most practical purposes, you can take the acceleration of the object to be zero until you're very close, though. But it is there.