Outside lamp has ground, no ground, no metal box.
Definitely put a GFCI upstream on this circuit. That will (barring a failure of the GFCI) protect humans from electrocution regardless of grounding.

Having unsheathed wires running through that frame and poking out for the fixtures could be a problem, though. Is that NM cable, or just a couple of single wires? The wires are supposed to be in a manufactured sheath (NM cable) or conduit, and I'm pretty sure holes drilled in a wooden frame don't count as conduit.

Can you pull new NM cable through the frame to include a ground wire?

Don't just screw a light fixture straight down to the wood frame. If you screw pancake boxes down to the frame (pictured), then you can attach the lighting fixtures to the pancake boxes in the normal way. If you can get a grounding wire in there, you would ground the box, and also ground the lighting fixture if it has a separate green grounding wire. The metal base of the fixture would also be grounded through the screw connection to the pancake box.

You also need to use a an appropriate cable clamp. One option is pictured, suitable for NM cable, and you would clip this into the pancake box from the back side so that the whole thing would still be flat enough to screw down to that wood frame, without having to auger the holes out bigger.