Chrome does not currently have any general-purpose APIs to support apps publishing mDNS records or listening for mDNS advertisements.
As for resolving mDNS names (names ending in ".local"), the general behavior is that if such a name is requested, Chrome will always delegate host resolution to the OS to allow it to be resolved via mDNS if and only if that is the OS's resolution behavior. So the ability for Chrome to resolve mDNS names depends on the OS on which Chrome is running and its configuration and installed libraries.