The docker CLI doesn't have functionality to search a registry, but you can use the registry's REST API. Assuming you're using the registry:2 image, then you can list all the repositories using the catalog endpoint:
$ curl https://my-registry:5000/v2/_catalog
{"repositories":["busybox","redis","ubuntu"]}
And then you can query the tags for a repository:
$ curl https://my-registry:5000/v2/busybox/tags/list
{"name":"busybox","tags":["2","latest"]}
These two API are useful, and the rest of them can be accessed with docker CLI, Here's the full Registry API spec.