Yes, deer will eat hay, any kind of hay, if they're hungry enough. If they're not, they won't touch it, or they'll be very picky as to what they eat. Turkeys are the same way, but turkeys burn energy a lot faster and will need help in the form of nutritious food a lot sooner.
That's why deer will devour hay in the UP in February and ignore it in Wayland. Winter is hard for months at a time in the UP, in the GR area, winter rarely lasts, ie., deep snow and bitter cold, for more than a couple of weeks.
I wouldn't worry about wasting the money on them unless you have at least 2+ feet of snow with a hard crust that lasts for several weeks and bitter cold along with it. If they can dig through the snow, they'll continue to eat natural mast and left-over ag crops.
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