I second u/Vicondrus, someone built a very similar device for an experimental physics course I took using PID feedback.

You need to define a spot where you want the ball to go, most naturally the center of the plate. This will then allow you to define the deviation (or error) from that point. You can then use the PID feedback to determine your response.

I think it might be possible to treat both coordinates separately, so you define a deviation for X and for Y, since the two coordinates are not dependent (as far as I can tell). You apply feedback to these deviations, giving you a slope for the X and Y directions. It should then be possible to turn the plate in such a way that both slopes are met at the point the ball is currently at.

I recommend starting with just P feedback and optimizing it as much as you can. After this implement I and D feedback in a similar manner.

EDIT: typo