HELP! Circuit board

If it is not grounded through the harness, then no.
Is there supposed to be a ground wire directly from the housing to the car?

I was thinking that, but then I see no place for a ground and all the other lights are working, including the left indicator and the high beam indicator- which are on the same board.

Absolutely ! If the gauge housing isn't chassis grounded the voltage limiter will seek a ground , fry a board, some gauges, self destruct.
I have a board here I'm repairing for another menber that experienced just that. He fryed more than this one board while playing with a panel out of the dash and hanging from the wires.
And as far as seeing no place for a ground wire... very little of these cars has a ground wire. Its mostly and eventually ALL chassis grounded through the metal bolts and screws.