Illinois does not require a lawyer to have insurance. However, the ARDC requests that you inform it each year whether or not you have insurance.
My advice to the public is that you should never hire a lawyer who is uninsured. First, if something goes wrong, you have little chance of any recovery. Second, the insurance process forces lawyers to look at their own conduct and consider whether certain practices are appropriate. Third, most policies have coverage for any grievance that someone may file with the bar association or attorney regulator. Having insurance gives you access to free CLE and you can usually call your insurer and ask questions about some practices. (Not every policy has all of these benefits, but your policy might have them.)
Most important of all, having insurance will discourage you from filing a lawsuit for legal fees against a former client.