With the horizontal wire spool position that is one weird-looking wirefeed unit, it looks like an Italian espresso coffee machine!
I have not found a circuit diagram for this machine on line; do you have any documentation for it?
I have found a service manual for the Snap-on YA212A welder.
It sounds like the output semiconductor which controls power to the wirefeed motor has gone short-circuit. This device could be a power transistor, an FET, or a thyristor ( SCR ). If you could provide a photo of the printed circuit board inside the feeder, and the part numbers of any large semiconductors mounted on heatsinks, it may be possible to guess at how the circuit works and which components could be faulty.
The Century board looks fairly simple. The fact that the drive transistor is available as a spare implies that maybe it is a common failure.
I have found conflicting data about the number of the transistor. It could be an NTE244, which is a PNP power transistor. However, it may be an MJ15015, which is NPN - the opposite polarity. Maybe you could check on your own board.
If the power transistor has failed and it is making the feed motor run continuously, then it will probably have a Collector-to-Emitter short-circuit. To test it, you will need to unsolder it from the board, and then use the "Diode Test" range on a digital multimeter, as per this procedure:
tore apart welder, it has 2 transistors, one on the board a MJ900 (failed base to emitter 1.9 and collector to emitter .467 and OL) and the second a MJ15015 mounted to a aluminum channel iron on the underside of the floor where the wire spool goes (within range on all tests except on the collector to emitter test continuity beep both ways)
The two power transistors that you found in your welder explain why I discovered data about two alternative components - there are in fact two different ones used in the machine.
The MJ900 is the equivalent of the NTE244, and it is a PNP Darlington Pair. That is a pair of transistors in one case, with the amplified current of one feeding into the base of the other, so they switch easily and quickly. The normal digital meter test procedure may not show the typical results on a Darlington Pair.
The MJ900 is rated at 80 volts, 8 Amps, and 90 Watts. The MJ15015 specs are: 120 volts, 15 Amps, and 180 Watts. Either of these two devices could easily run a typical wirefeed motor, at 24 or 48 volts and 2 or 3 Amps. It is possible that one transistor is in series with the motor to control the speed, and the other is across the motor terminals ( i.e. in parallel with it ). This second component will get switched on when the motor is not under power, to short it and bring it to a rapid halt, preventing run-on of the MIG wire spool. You would need to trace the motor wiring to confirm this. To cause the motor to run continuously, the transistor which is in series with it will be short-circuit from emitter to collector, or incorrectly constantly turned on by a signal on the base.
Replaced both transistors but wire feed, etc engages once and a while when squeezing trigger on gun. Wire feed works when inch button is depressed and speed changes as dial is turned, the gas solenoid works when the purge button is depressed. Unhooked leads to gun switch and did continuity test - passed and #37 on the ya240a diagram if the same on the ya240 (thermostat) had continuity, the fan doesn't run.
It sounds like you have made good progress, in that the original fault of the wirefeed motor running constantly at a fixed speed has been solved. Now the torch trigger switch is intermittent in operation. The local controls - the wire inch button and the gas solenoid purge button - do work, so there must be a sporadic failure in the trigger switch circuit. Without a circuit diagram, you would have to check for loose connections and bad soldered joints. The torch switch often operates a small relay, on or near the printed circuit board, which in turn drives a contactor ( a heavy-duty relay ). Sticking relays or dirty or burnt contacts could cause intermittent faults. Sometimes electronic devices such as TRIACs ( a semiconductor AC power switch ) are used instead of relays.
Apart from wire feed and gas, the trigger switch should also be signalling the main welding power source to turn on. Does this happen reliably, or intermittently, or not at all?