When did Illinois begin requiring gun registration?

In 1838 Georgia banned all handguns. Supreme Court found the ban to be unconstitutional.

However, to further answer the question, contrary to popular belief, there is very limited registration in the U.S. A few individual states require it, but only a few. There is no federal registration of "normal" firearms, such as handguns and rifles. Some items, known and NFA or Class III items must be registered with the federal government, and a few states don't allow those items within the state. This includes items such as machine guns, silencers, short rifles, and short shotguns.