And the award goes too.... DM! RFC 164 from 19 May 1971 says DM was to have NCP by end of week. RFC 211 certainly lists DMCG among other hosts. RFC 148 seems to imply NFP was working to some extent even earlier.
RFC 342 from 15 May 1972 adds AI as a Network User. In RFC 344 is had become a server. RFC 366 from 28 June 1972 says "During this period the MIT Math Lab PDP-10 (Network address 198) became a server."