Maybe the 10 and 100 relays are stuck on? A quick way to check this is to play until the no-scoring problem occurs, then look at the score, then pull the plug. If the 10 and/or 100 score reels step up as you pull the plug, that means that their coils were already energized.

After that, you could take off the playfield glass and try working switches by hand. For example, reset the game and leave the ball in the runway, then hit 10-point switches enough times to carry over to the 100s. Try each and every 10 point switch. When you get one that doesn't score, reset the game and do the procedure again, but hit the suspect switch first and see how many times you can hit it before scoring stops. Similarly, focus on the spinner, bumpers, 500 point lanes, etc. in separate tests.

You could also go into the backbox and try working the L, M, and N relays by hand by just pushing the plastic insulator (with all the switch leaves sticking through it) toward the coil. Each relay should only stay on long enough to step its associated scoring reel coil. Try it on each of the four players.
.................David Marston

That's probably relay M, the 100 point relay. As I wrote before, it should only stay on long enough to step the 100 point scoring reel coil. But you have four such coils, one for each player. So try starting a four-player game, leave the ball in the runway, hit some 100 point switches by hand (or the spinner), and watch for that relay to get stuck on (easier if you have 2 people, one in front and one in back). Then pick the ball up out of the runway and drop it directly into the drain so that the game can advance to player 2, hit some 100 point switches again and see if the relay sticks. Repeat that step for players 3 and 4. Don't do 500s if you can make the problem occur with 100s, as it would just add more factors to consider.
.................David Marston

Actually, it's only the scoring reel that has to complete its end of stroke. The center chime is energized (plunger up) anytime that the M relay is on.

Like HowardR I suspect the scoring reel end-of-stroke. I notice that players 2-3-4 are never getting points in your tests, but here is a new variation of the test where we avoid using player 1:
Try starting a four-player game, leave the ball in the runway, and for player 1, score only 1000s by hand, such as from the Vari-Target.
Then pick the ball up out of the runway and drop it directly into the drain so that the game can advance to player 2, hit some 100 point switches and see if the relay sticks. Repeat that step for players 3 and 4.

If relay M and the scoring reels interact properly for the other 3 players, then the suspicion points toward player 1 scoring reel. If Player 2 has trouble with the 100 point scoring, you can re-run the above experiment but score nothing but 1000s for both players 1 and 2, and see if Player 3 has trouble with its 100 point scoring when it is the first player to make use of the M relay. Likewise, you could avoid 100-point scoring on all but Player 4.

If more than 1 player has the issue, then we have to ask why the M relay is not dropping off when the scoring reel (of whichever "bad" player) is pulled all the way in. There could be a short circuit somewhere.
.................David Marston

Also look all around relay M for any stray wires that may be brushing up against a switch leaf or the non-black wire on the coil. The problem can be described as the M relay getting power more than (or "for longer than") it should be.

As I mentioned in post #2, you can work relay L by hand by just pushing the plastic insulator (with all the switch leaves sticking through it) toward the coil. Since L is not a suspect, it would give you a better sense of what is supposed to happen. Then compare to activating M by hand.
.................David Marston

When you get the schematic, a good place to start is to look at the 6-volt lighting zone. That should get you accustomed to the symbols quickly.

Then find the M coil (using the coil index) and look at all the things that activate it, particularly how it holds itself on until the scoring reel reaches end-of-stroke.
.................David Marston