The connector appears to be a Molex 0.156.
A similar post can be found here:
I have to replace a few molex connectors for a gi issue Where do you guys get these I will need the pins as well Thanks all
www.pinballrevolution.com
I bought many Molex pins & connectors from Great Plains Electronics in the past, great parts & service.
With regards to the lower ball kicker, the issue is unlikely to be the coil itself.
Probably more with its drive or its switch detection, e.g. either its switch is not detected or is misadjusted,
there is a broken wire, bad connector/pin/crimp, cold solder joint on driver board (header) or a bad drive transistor.
To help pinpoint the issue for this, you can go in switch test and solenoid test and see what is not detected or not working.
The game manual has instructions on how to enter these test modes.
UPDATE: 'lower ball kicker' - is this the outhole? If so, make sure there are two balls and that they are detected by the switches.
Are the drop targets resetting at game start? If no solenoids work, could also be a shorted drive transistor, burned coil and/or burned 2.5A solenoid fuse.
Good luck!