I changed the headlamps on my wagon and noticed the driver side lamp wasn’t working. I took it apart again and noticed a yellow wire that goes from the lamp to a white plastic connector had broken at the connector. How do these wire connect? The side the wire is on is flat. The other side has pins that has another multi pin connector attached. Do I have to buy a new harness or can it be repaired with solder?

If the wire isn't inside the headlight assembly at the bulb, I'd sure try solder or a crimp splice and some heat shrink tubing insulation. If it is inside the headlight near the bulb, it may be subjected to higher temps and may need to have a new socket spliced in.