Can an employer suspend you for five days without pay for something another employee said you did? Without hearing your side of the story.
Generally, yes. Most employees are at-will, meaning they are not subject to a formal employment contract. The employee can leave when he or she pleases and the employer can fire / suspend as he or she pleases. An employee of a private employer (versus a public employer, like the state or a city) is not subject to constitutional provisions like "due process" and does not have to provide any such process before disciplining an employee.
The exceptions to this would be (1) if you are a public employee or (2) if you had a formal contract that specified disciplinary procedures before discipline. If you are unsure, or want to ask more questions, I would advise speaking to a lawyer of your choice through the Illinois State Bar Association: www.illinoislawyerfinder.com. Search for "employment" attorneys, not "labor" attorneys.
NOTE: This answer is not intended to be legal advice and should not be construed in that way. This answer does not create an attorney-client relationship and no such relationship may be created absent a signed retainer agreement. The author is licensed in Illinois only, and his answer is for educational purposes alone.