Illinois, Northern District Petitions for Naturalization, 1906-1991.
There are two things you should know about the title.
First, it says "petitions," but the database actually includes declarations and certificates of arrival, too.
And second, it says "1906-1991, but the images are being added and the collection isn't yet complete.
Current coverage (updated 1 May 2014): First record #98951 from 1931; last record #278950 from 1943.
So, how do you find these records? It's pretty simple.
First, search for names in the Illinois, Northern District Naturalization Index, 1840-1950
When you find a match, check the "Title and Location of Court" box. If it says, "District" (or U.S. Dist.") then there's a chance you can find the matching record online.
Check the naturalization date. If it's between 1922 and 1940, look for the petition number. It will be found in the "Certificate Number" box and it's usually prefixed with the letter "P."
Next, go to Illinois, Northern District Petitions for Naturalization, 1906-1991, click "Browse through images" and click on the "Cook County" link.
The numbers you see are petition numbers. If you can find a group that includes the P-Number from your Soundex card, you're in luck.
The online images will include certificates of arrival and declarations of intention in addition to the petitions, but they will be arranged numerically by petition number (even if the petition number doesn't appear on the certificates or declarations).
And what if the record you need isn't online yet? If it's from the District Court, I'd suggest submitting a request to the National Archives at Chicago online. The cost is $10 and I hear the turnaround time is pretty quick.
If it's from any other Cook County court (Circuit, Superior, County, or Criminal) then it will be at the Cook County Circuit Court Archives and I'd suggest having someone pick the record up in person. It's quicker and less expensive than by mail and it opens up the possibility of having color scans or photos instead of black and white photocopies. You can connect with a few researchers who offer that service by visiting the Circuit Court Archives page on our Genlighten website.