The problem you will have is that modern day switches will stop you sniffing traffic destined for specific MAC addresses so if its hard-wired then you are going to run into trouble. If on the other-hand its a unsecured WiFi connection, or a network with hubs (rare now!) rather than switches then you may have a chance.
You'd need to switch the ethernet controller into promiscuous mode and listen for the DHCP REQUEST and DHCP OFFER commands, inspect the contents and see whats going on.
Another method would be to listen for ARP and RARP requests on the network and use a hardware lookup table for the MAC id to determine who makes the chip-sets, its not going to give you awesome accuracy since a lot of chips are re-used on different devices but it may give you a little bit more info.
In terms of the DHCPd server, unless you have access to that physical machine I'm afraid your probably out of luck, DHCP is a designed as a offering protocol, not a querying protocol.