Can an OpenVPN server be set up on a machine with a single NIC?

Absolutely, as long as you make sure that:

The NAT gateway on the server's network has a port forward rule for TCP/UDP 1194 to the internal address of the OpenVPN server machine.
If you are using routing rather than ethernet bridging mode and would like connecting clients to see the whole LAN rather than only the server machine itself, you need to add an internal LAN route to the LAN gateway so that the private OpenVPN subnet (declared in theserver, ifconfig, or ifconfig-pool directives) is routed to the OpenVPN server machine (i.e. its internal address).