Florida does at least...I'm surprised other schools with an online component to their supplemental applications/required stuff do not. At least not that I've found. And it is so hard to get through to the admissions offices by phone to confirm.
It might be helpful to provide links in your post to places that have online status checking!
Colorado State University Admissions Portal - Kept up to date and very informative. Has current admissions status, status history, submitted documents, and residency status.