What is the material property that controls the amplitude of the magnetic field oscillation? What is the material property that controls the amplitude of the magnetic field oscillation in an EM Wave and how is it different than the material property for electric fields oscillation?
 A: Your question is a bit strange. EM waves mostly associated with their travel in vacuum. Vacuum is not a material and by this hasn’t material properties.
But where you are right, even a vacuum isn’t empty. It is “filled” with a gravitational field and also with electric and magnetic fields.
Using the approximation of an uniform distribution of these fields (usually saying “in empty space”) an EM wave has unique characteristics:


*

*the direction of the electric field component is perpendicular to the direction of the magnetic field component and both components are perpendicular to the direction of propagation

*the relation of the intensities of the components is a constant value

*and last not least the speed of propagation (the speed of light) is a constant.




The classical behaviour of the electromagnetic field is described by Maxwell's equations, which predict that the speed c with which electromagnetic waves (such as light) propagate through the vacuum is related to the distributed capacitance and inductance of the vacuum, otherwise respectively known as the electric constant ε0 and the magnetic constant μ0, by the equation 
    $$c={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}$$


The quote is from Wikipedia and if we exchange electric constant by vacuum permittivity and magnetic constant by vacuum permeability, your questions may be answered for a vacuum. 
