One half of the board takes full mains voltage, transforms it down to low voltage...rectifies it and then the trigger switches it to pull in the relay.
The relay switches the mains in to the big transformer, turning the welding current on.
My bet is that the main transformer has a little winding (along with the big high current ones for welding from) that produces a low voltage to drive the wire feed motor.
The other half of the board, I believe, has an SCR or thyristor circuit to control the motor speed.
If the relay isn't pulling in when you squeeze the trigger, then either the on-board transformer or the bridge rectifier may be shot.....
"Life isn't about waiting for the storm to pass... It is about learning how to dance in the rain"