There was a day in December or January that was super windy here, and I remember being outside and noticing the wind was blowing and gusting from multiple directions. I thought to myself, "this would be great tornado weather if it wasn't so cold", believing exactly that- that tornadoes only happen in warmer temps.
Then I heard in the news the following day that there were indeed tornadoes out in the middle of the state (the boonies). I was obviously mistaken in thinking the airtemp needed to be higher, and they can indeed occur in winter. I'd imagine that type of vacuum pressure and wind speed would have to pick up snow if there was any on the ground, even if it were hard-frozen and crusted. A tornado while it was snowing/blizzarding would be something else entirely.
It isn't going to look like white snow once it's in the 'nado, though. It'll just become dirty water and/or debris like the rest of the things carried by tornadoes.
Edit: yep, I saw a few of those snow-devils that
@Red Squirrel mentioned on that day referenced above. There was enough loose powder on the ground that I spotted two little ones maybe 2-3m high, real thin and wispy and didn't last more than 10 seconds.