Yes there is a lottery for Turkey hunting permits here in Michigan. You apply by the zone/area and dates you want to hunt. First date is some time in April and the last is late May.
Your limited to a week last time I checked.
Some times you don't get a permit for what you applied to but can reapply for a any zone any date drawing.
I stopped turkey hunting when a guy from Illinois stopped and asked about hunting turkeys on our farm where I had applied but on the second chance I drew for a area 3 hours across the state.
I felt I should have first choice on my home property and the person from out of state can get what ever is left.

I just raise a turkey or buy one from the farmer down the road.

Al