Do I need to print and mail my Illinois tax return if I am doing it for the first time in the state and I don't have an Illinois Driver's License?

Yes.  IL wants a state issued ID or drivers license in order for you to e-file your IL return.  If you do not have an ID you need to mail in your return.

Thank you so much. I really appreciate your prompt response. 

You are welcome.  When you mail a tax return it is a good idea to use a mailing service with tracking like UPS or certified mail so you know they received it.