You say "this time" - do you mean the flashers went bad before and moving the connector fixed those exact flashers? If so, the connector probably needs to be repinned and/or solder reflowed.

Otherwise it sounds like a loose or broken wire, or a bad transistor.

Have you:
checked fuses
replaced one of the bad flasher bulbs with a known good bulb
checked for power at the flasher socket with a DMM
checked continuity on the flasher wires from connector all the way down the line to see if there is a short or loose wire
checked transistors driving the flashers

Probably not but I don’t know. There may be a card in the backbox that says. Otherwise look in the manual.

If there is power to the socket and there is continuity on both wires from the flasher to the connector it’s probably a transistor.

If you have continuity from all the flasher sockets to the connector at the board, and power at all flasher sockets it /has/ to be upstream. That would likely mean a transistor or a pre-driver.

I'm not sure that's a good idea.

On system 11 games (which this is /not/), you have to either cut certain resistors or cut ground to a certain board to use LED flashers, because the resistors. I don't know if these Data East machines are similar. Otherwise that resistor isn't optional.

That could possibly be it. If you're not comfortable with board repairs, you should send it out to a pro if you don't have one local. They can check the rest of the board while they have it.