If you want to use the Matcaps that come with blender in your renders, you can add them as an HDRI. Go to your shader tab, select "World" on the left side (blocked by the file manager window in my image - switch it from "Object"), and add an Environment Texture. Click the choose image button, and navigate to
\2.90\datafiles\studiolights\matcap\ (or whatever version you're using). The files (.exr) are all in that folder. Choose the one you like.
This will provide the lighting and reflections from the image, but will also make it display as a background as well. If you don't want the background image component, just mis it with a regular background, using the Is Camera Ray output of a Light Path node as a mix factor.