is it ok to ship ammo to a person in cook county?

There are no Cook County restrictions on ammunition, but many online vendors won't ship there or will only ship to an FFL.  Their businesses, their rules.

The only restriction is for the entire state. The IL buyer needs to possess a valid FOID card.  Illinois law says the buyer is required to provide a copy of the FOID and DL to any seller, either in state or out of state.

(430 ILCS 65/3)(b-5) Any resident may purchase ammunition from a person within or outside of Illinois if shipment is by United States mail or by a private express carrier authorized by federal law to ship ammunition. Any resident purchasing ammunition within or outside the State of Illinois must provide the seller with a copy of his or her valid Firearm Owner's Identification Card or valid concealed carry license and either his or her Illinois driver's license or Illinois State Identification Card prior to the shipment of the ammunition. The ammunition may be shipped only to an address on either of those 2 documents.