Technically yes, you can practice in CA without a JD, but I have never known anyone like that. There may not even be a single person who practices without a JD anywhere- it really isn't an option. An LLM is a masters after your JD that is a specialization. Most people use them to boot their resumes when the go to a less prestigious law school. Bachman, for example, got her LLM in tax law at W&L.
IL has a bunch of schools. If money isn't a problem you can look into part time programs and commute a day or two a week. The problem with online schools is that the ABA and all states but CA don't recognize them. Its analogous to a PhD or MD from an online school- kind of useless. Online grads can't get licensed outside of CA.
You don't need any special certification to practice immigration or anything else for that matter, just a JD. The word "certification" actually makes me a little suspicious. Lawyers specialize with seminars and LLMs for resume purposes, not certifications. You can check the distance education boards on LSD if you want, but "online law school" is somewhat of an oxymoron. I can understand why some people go to state accredited schools, rather than ABA. Online school students, however, seem like they are getting bamboozled. Bottom line, you'll have to find a brick and mortar school if you want to practice law.
List of IL law school-
http://www.alllaw.com/state_resources/illinois/law_schools/Good luck.
P.S.- Don't tell the online law students on LSD I said that. They're a pretty darn defensive group. I don't want to be inundated with comments by people defending such hopeless position of "online law student"

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