First, try contact Legal Aid, such as Cabrini Green Legal Aid. Removal is fact specific, and there's little that an online forum can assist with. Read 750 ILCS 5/609 (The removal statute) to see what the court is required to look at, and weigh that against the specific facts of your case.
Again, call every legal aid in the area to see if anyone can help you first before handling this on your own. God speed.
Removal is a complicated legal process which you should not attempt without help. The fact that you do not know where your ex is may make things a little easier for you, but if you have not properly served him by publication, your motion will probably be continued or denied outright with instructions to serve your ex. You really need to discuss your case with an experienced attorney ASAP. Contact your local legal aid office or law school clinic if you qualify for free services.
You are in over your head. No court will allow you to remove the children unless he is served with the Petition.
Some of the informaton you reported is helpful but it is not enough for a removal. Try Illinois Legal Aid Online which will help yuo with forms and locating Legal Aid officed and Legal Clinics.
You should look at the case entitled In Re the Marriage of Eckert (119 Ill. 2d 316). This case discussed removal of a minor child in great detail and set out a number of factors that the Court should be considering when making its decision. The more arguments you can present to the Court that are in line with the factors presented in the Eckert case, the stronger your case will be. There is another case, In Re the Marraige of Collingbourne, that is more recent and expands upon the Eckert factors, so you should read this case, as well.
If you have any other questions, you should consult with Legal Aid or request a consultation with an attorney prior to your Court appearance.
Sign up to receive a 3-part series of useful information and advice about child custody law.