If you have a geometry where you can use image charges then nothing is changed by replacing the point charge with a distribution of charges. Electromagnetics follows linear superposition and since this is a static problem your total solution will be the solution for each individual charge in your distribution. So you can just find the image of the charge distribution and calculate the field from the image.
Of course using images is only limited to a small subset of problems, but the point still stands. If you can solve for a single charge then you can just extend the method for a distribution of charges. The primary difference will lie in that you will have to integrate over the charge distribution (much like you would sum up over a finite set of charges) to get the total field.
This is assuming that your charge distribution will not change in the presence of an applied field. But if you were to charge a conducting disc then the charge distribution on your disc will change due to the interaction of the original charge distribution with the induced charge distribution on your conductor. In this case it is a much more difficult problem. However, you can solve this problem more easily by first noting that a conductor is an equipotential surface. So instead of trying to work from the standpoint of:
I have charge distribution A on a conducting object. I bring in conducting object B. What is the new charge distributions on A and B?
You work from the viewpoint of:
I have an object A with a potential of X. I bring in B. What is the charge distribution on A and B where A has a potential of X and B has a potential of 0 when they are infinitely separated?
You can represent this problem as a differential equation, Laplace's equation, and using the appropriate boundary conditions you can then solve for the potential on the surfaces of the objects. There are some problems where you can solve this analytically (method of images is just an example of this) but sometimes you need to use numerical techniques like method of moments or relaxation methods.