Is this article implying that a non-resident of Illinois is not allowed to buy ammunition in that state?
The law is sort of wierd. If I remember right, it reads that you must have a valid foid to purchase or posess firearms or ammunition, the exception being if you hold a valid non resident hunting permit. There are also exceptions for non-residents traveling with firearms and for an FFL selling to an out of state resident (for example in a border town). Illinois has reciprocity agreements aith all the bordering states for non resident purchases of long guns (and probably ammo) but if you are not from a bordering state you are out of luck.
Illinois weapons laws are so confusing that most law enforcement officers dont even properly understand them.