It should run fine sans LCD. Just hook it up. You could get an inexpensive LCD monitor and build your own imac. Is the problem the logic board flaw that those are notorious for? It doesn't sound like it as if it were it wouldn't output any video at all. It may just be the cable running from the board to the screen. Not a hard fix.
PB G4 15" 1.2ghz - Pismo G3 400 mhz - iBook G3 500 "Jukebox"
"SawSmurf" G4 500 - Quicksilver G4 Dual 1ghz modding...