We have a craftsman snow thrower model #536.881112. It starts up but will not engage to move.

Check your traction cable , or drive belt
Does the auger turn when engaged ?
Luck

Take a close look at the exposed end of the axle showing through the wheel, does the axle turn but not the wheel when you try to engage?

The Auger does move. And John said he will check the manual for the location of the traction cable and drive belt and will look to see if they are intact and doing what they should be doing and we will get back to you, thank you.

Bob has a good point some of these have pins in the axle that you move so they can free wheel.
Luck