Unfortunately, state docket searching is very spotty and no service has complete coverage. In many rural counties there aren't even electronic dockets online, requiring instead a trip to the courthouse.
Bloomberg Law is the most efficient to search, but only has 11 counties covered.
Westlaw only has 8 counties, but two of them are not on Bloomberg. They also cover Montgomery County, but it's archival-only, so new cases won't appear there.
Lexis is not transparent about its coverage. They have Illinois Circuit Court dockets but don't say which. However, we have access to their CourtLink service, which shows 8 counties; however, 2 of them are out of date and none of them are unique to CourtLink (i.e., they're all on either Bloomberg or Westlaw).
Finally, one of the larger counties not covered by any of the three, McHenry County, does have electronic court record searching availability on the clerk's website so, unfortunately, the fact that electronic searching is available does not mean they're on one of the commercial services. Thus the other county clerk websites will have to be checked in order to be comprehensive (assuming remote electronic searching is available, which it may not be). Illinois has 102 counties.