My adult son is mentally ill and currently in a hospital in the Chicago area. I live in Florida. How do I apply for guardianship from another state?

Have you tried moving your son without a guardianship? Assuming your son would be willing to move, and the hospital would release him to your care, you would only need to file for guardianship in Florida. If your son will not move to Florida or the hospital will not release him to you, you will need to contact an attorney who practices guardianship in Chicago.  You may need to file for a guardianship in Chicago to move him to Florida, and then contact a Florida attorney to transfer the guardianship to a Florida court.