I want to know if a felon can become a licensed cosmetologist in the state of Illinois

Here is the application and requirement info from the State of Illinois:
http://www.idfpr.com/renewals/apply/Forms/COS-EX.pdf

It does not specifically state anything about previous felony convictions.

You should, however, contact them and ask them. Here is their contact information:

Illinois State Board of Cosmetology Contact Information
Illinois Dept. of Professional Regulation (http://www.idfpr.com/)
320W. Washington St. 3rd Fl.
Springfield, IL 62786
217-782-8556 (phone)
217-782-7645 (fax)