You can get an Illinois broker license if you are not a state resident. However, you must have a Social Security number or an individual taxpayer ID number.
You can find the rules regarding real estate in the Illinois General Assembly under Professions, Occupations, and Business Operations (225 ILCS 454) Real Estate License Act of 2000. The administrative code is found under title 16, chapter 8, subchapter B, part 1450.