Check? As in the ability of the police to see if electronic records indicate that a license in valid? I'm not 100% sure... The Delaware Justice Information System (DELJIS) can look up many things, but I don't know if a CCDW is part of that. You may want to ask over at Delaware Open Carry: Delaware Open Carry
In lieu of keeping a public record, Delaware residents are required to public their full name and home address in a news paper that carries at least 35% of the circulation in their ZIP code. I personally don't know if they store the information in DELJIS as well.