I had this issue myself on my William's Wild Card. Here's what I did.

1: While I didn't need to do this, check the fuse holders and make sure the fuses are making good contact. One good way is to lightly pinch the clips closer so it's getting a firmer connection.

2: Most likely the issue is the jones plugs. Find the jones plugs that go from the playfield to the bottom board. Remove them, and clean then with either a bristle brush or 600 grit sandpaper. Personally I prefer sandpaper. Afterwards I usually wipe them down with 99% isopropyl alcohol though this isn't needed. If you do just make sure it's at LEAST over 90%. I find anything less leaves a white residue. I'd do this to all the jones plugs, good preventative maintenance.

Also as a note, if you use alcohol I'd wait at least 30 minutes to and hour before turning the game back on just be sure the fumes are all evaporated. While I've never had an issue with the fumes catching fire even after 5-10 minutes, better safe then sorry as they say.

3: Finally you wanna clean the female ends of the jones plugs. What I like to do it take some hard bristle pipe cleaners, dip an end in 99% isopropyl alcohol and use that to clean each of the female connectors. Afterwords I take some canned air to quickly remove any remaining dust/gunk/bristles.

After a good cleaning I've had 0 lighting and connection issues. I had to do it twice myself, so do it a couple times if it doesn't work the first.

If that doesn't work I'd grab some schematics and try and see if there's a switch that provides power to the lights. I know on gottliebs the tilt relay will kill all the lights on the game if you tilt, but I don't think williams do that.