i'm an electrician, not an electronics guy. i googled mig welder electrical diagram and came up with this schematic (page 36 on the manual pdf link). a magic board controls both the arc and the wire feed motor. the simple way would be to interrupt the ground with a relay and a foot pedal.
my way would be to use a relay and a timer: so that when you let of the trigger a timer would kick in for a determined time and power the feed rollers. set for a second and you would have a seconds worth of wire stuck in the puddle, set to zero and it would be a welder. use a time delay relay like
my bigger concern would be the amount of heat and puddle required to make enough puddle to stick the wire to it.
when i did automated welders we had timers for everything even a timer to reverse the wire feed to pull the wire back for the operator. but those were 600 amp continuous power supplies with separate wire feeds running 7/32 flux core wire at 1 ft per second. now that is some serious welding. 55 feet in 56 seconds to weld the backing plate bracket on a ford super duty axle.