Here you go.
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If a non-resident is coming to Illinois to hunt and would like to bring their firearm, how do they legally transport it?
Non- residents must be legally eligible to possess or acquire firearms and ammunition in their state of residence. It is recommended that, in order to be in compliance with all statutes, non-residents transport all firearms:
Unloaded, and
Enclosed in a case, and
Not immediately accessible or broken down in a nonfunctioning state.
Best to do is at border take firearm out of your holster, remove mag and remove loaded round from the chamber, Lock the firearm in a "container" (Can be a case, the shipping container, etc.) this "container" must zip up, or lock up someway that the firearm can't fall out. And it must be kept in the trunk of the vehicle. (or the farthest point in the vehicle from anyone.)
Is it legal to have ammunition in the case with the firearm?
Yes, so long as the firearm is unloaded and properly enclosed in a case.
I have a friend/relative who has a "conceal and carry" permit issued in the state in which they reside. Is the permit recognized in Illinois?
No. Non-residents are subject to Illinois law, restrictions, and penalties, and should be familiar with them if the non-resident plans to bring a firearm into the state of Illinois.