Spanish Store Chicago
The biggest I know for all Hispanic products is Tony's Finer Foods, which has several branches. I don't know whether they stock a lot of Spanish products qua from Spain but the long aisles of Hispanic items would definitely be worth exploring. I know the stores at Fullerton & Central Park and Hamlin & Elston are pretty big and the one in Niles near Greenwood & Dempster is gargantuan.
This thread is old but addresses the issue: http://chowhound.chow.com/topics/511388
I'm not aware of a dedicated Spanish store that has opened since then. I was in Fox & Obel today, looking for other things, and noticed Arroz de Valencia in a sack just below a shelf of some fish (can't recall exactly which) in oil. If, in fact, I stumbled on a Spanish section, it was a small one. On a related note, I next went to Publican Quality Meats, and they were selling Avec's chorizo-stuffed Medjool dates with smoked bacon, $1.50 each, stuffed and ready to bake, and their excellent piquillo pepper-tomato sauce.