Since you're on the same subnet, you can P/Invoke GetIpNetTable to get the webserver's ARP table. If you do this real-time, no additional work would be necessary - since you're having a conversation with the client, you'll have the ARP info. Otherwise, you'd need to construct an ARP request or some IP traffic (say, a ping) to get it in the cache - and note that due to DHCP and other network vagaries (like a machine being turned off), it is possible that converting IP to MAC later will yield a different answer.
Note also that any external clients (ie., ones across a router) just won't show up in the table - so be prepared to deal with that as well. If you need a MAC for them for some reason, it's technically your router's MAC.