To be a snow plow driver commercially for a state you usually need a CDL drivers license to be hired as a state DOT snow plow driver and operate their snow plow trucks.
But if you want to slap a snow plow on your own truck then all you need is your regular drivers license to drive the truck and plow the snow for people.
However if you plan to plow snow for businesses they might require you have insurance and a business license for operating your snow plow business.
That is for protection of their business and protection of your business in the event that something happens.
My dad used to plow snow in his own personal vehicle and made some good money at it and he wasn't licensed but being licensed and insured as a snow plow business makes you look more professional and leads to more business.