The 125 C thermal cutout switch will normally be closed. It is probably glued to the side of the main transformer...? Tapping or hitting it may help it to close again. If it has failed open-circuit, you could bypass it by joining the two wires together. For a replacement, search eBay for "thermal cutout switch".
The small transformer on the board drives the relay when the torch trigger switch is pressed. The relay is marked "18v DC", so you need to search eBay for "PCB transformer 18v". To confirm that the component actually has failed, check for AC voltage across the secondary when the welder is plugged in and powered on ( and the thermal cutout is closed ), or for continuity of the primary and secondary windings. The resistance of the primary ( which is the most likely to have burnt out ) may be surprisingly high - up to 3k Ohms, and a couple of hundred Ohms for the secondary.
Post #3 of this earlier thread may be a circuit diagram of the welder:
Hi all! was using my welder yesterday and all of a sudden the wire feed stopped! I’ve hooked up the motor to a battery and it works with a 12v feed but when I try it with the machine wires there isn’t even a 1v feed. You can hear the relay click but no wire movement!