Amtrak used to have a ban on firearms even in checked luggage. That said I've read that quite a few bird hunters would travel frm Chicago to SD for their fall hunt.
It might be fun to take the Empire Builder from Chigago to Seattle, grouse in WI, pheasants in Western MN and ND, Sharptail, Pheasant and huns in Western ND and eastern MT Quail in central WA and Seaducks in western WA. You would have to book a guide at each place or at least rent a vehicle.
There used to be a high end bird safari by train in Africa, don't remember the details
