Why is there an induced EMF in a plastic ring?

An electromotive force doesn't require a conductor --- it doesn't even require matter. The electromagnetic field is a local property of the vacuum, governed by Maxwell's equations. The relevant one in this case is

$$ \vec\nabla \times \vec E = -\frac\partial{\partial t}\vec B $$

That is, at any point in space, a changing magnitude or direction for the magnetic field is inextricably associated with an electric field with nonzero curl. Because electromagnetism obeys all the symmetries of special relativity, it doesn't matter whether the point of interest is stationary and the field there is changing, or if the point of interest is moving through a region of static but nonuniform magnetic fields.

Now, a "conductor" is some material with the property of "always" having $\vec E=0$ inside. Since the changing magnetic field is associated with $\vec E \neq 0$, then the charges in the conductor where $\partial\vec B/\partial t \neq 0$ must move to produce $\vec E=0$ by superposition. You can think of that as motion due to the Lorentz force if you like. The Lorentz force,

$$ \vec F = q\left(\vec v\times\vec B + \vec E \right), $$

implies that a charge must be moving to experience a force from a static magnetic field. However a changing magnetic field produces a nonzero $\vec E$, and can therefore exert force on stationary charges.