According to the following documentation it should be possible to integrate clair with a docker private registry. How to do that?
Clair can be integrated directly into a container registry such that the registry is responsible for interacting with Clair on behalf of the user. This type of setup avoids the manual scanning of images and creates a sensible location to which Clair's vulnerability notifications can be propagated. The registry can also be used for authorization to avoid sharing vulnerability information about images to which one might not have access.