indicates that Illinois prohibits machineguns:
It is unlawful to sell, manufacture, purchase, possess or carry any weapon from which more than one shot may be discharged by a single function of the trigger, including the frame or receiver of any such weapon.
It is unlawful to sell, manufacture, purchase, possess or arry any combination of parts designed or intended for use in converting any weapon into a machine gun, or any combination or parts from which a machine gun can be assembled.
There is an exception to the above prohibition, providing for the manufacture and sale of machine guns to law enforcement personnel only. This allows class 3 dealers to possess machine guns as long as they are for distribution only to law enforcement and military personnel provided such weapons are "broken down in a non-functioning state or not immediately accessible."
This prohibition does not apply to persons licensed to manufacture machine guns or ammunition under federal law and who are actually engaged in the business of manufacturing such weapons or ammunition, but only with respect to activities which are in the lawful scope of such business, such as the manufacture, transportation or testing of such weapons or ammunition.
It is unlawful to possess a silencer.
James Bardwell's NFA FAQ (previous post) indicates the only Class III item you can own in Illinois is an AOW (Any Other Weapon), which limits you to generally amusing and useless things like short-barreled pistol-grip-only shotguns.
Motivated by the same reasons as you, and faced with the same restrictions, I got an AOW shotgun. It was an interesting exercise, but I suggest you seriously consider putting the money and effort into something more useful.