RE: Illinois Firearm Deer Applications...
From the regulations page 13, left side half way down....................
PERMIT LIMIT
No hunter may receive, or attempt to receive, more than one either sex firearm permit and one antlerless-only permit prior to the random daily drawing in September. Permits remaining after the lotteries are issued by random daily drawings until the county permit quotas are exhausted.
So you "Can't" apply for multiple permits (counties etc.) until the regular first and second lotteries are complete.