Welcome.
Technically, yes. If you're an electronics engineer you could re-design / modify the control circuit board to incorporate a relay so the torch trigger the controls the welding voltage as well as the wire feed. Or fit a universal PCB to replace the existing one.
In practice, no. It's not worth the effort on a very basic hobby set. Sell it and buy a known good model. I'd guess that you've bought a new-ish, AC gasless MIG? Or have an early 1980s Mightymig 100 or the Sureweld equivalent?
A MIG welder that is not live torch has an auxiliary source of power that comes up when the machine is initially switched on. A hobby-type machine will usually have a small plastic-encapsulated transformer mounted on the control printed circuit board. The 12 or 24 volts output is rectified and used to control the AC mains input power of the main welding transformer, via the trigger switch in the MIG torch. It does this with a relay on the PCB, or an electronic switch - a Triac ( which would also be on the PCB, mounted on a heatsink ). Thus the torch trigger switch controls not only the wirefeed motor but also the welding power.
You could check the PCB in your machine to confirm that the components I referred to above are missing ( as opposed to being defective - e.g. a relay with contacts that have stuck closed ). What model of MIG welder do you have? A photo of the data plate would help to determine if, as a111r suggested, it is worth modifying. What work do you intend to use the welder for?
Contactor - big one , then cut the earth ( return work lead ) and put that either side in the contactor , this is how the early cebora / snap on did it