You live in Illinois. Illinois register gun owners (via FOID); Illinois does
not register guns.
If you were buying a gun from an IL gun shop (an FFL) then you would need to fill out the Form 4473... but this is a federal (BATF) form,
not an Illinois form. But you are not doing this; you own an old rifle passed down from a family member. It was a private transfer, and the transfer has already taken place.
The only other scenario is if you live in a town, township or county in Illinois that requires gun registration. Then you need to comply with any local laws.
Here's a
link to a brochure by the IL State Police regarding private sales and transfers (the ISP are responsible for gun laws enforcement in IL). Just make sure you are in compliance. The rules are not hard to follow. Consider going back and creating a receipt for the transfer, if it will give you more peace of mind.