You can enter collections of bugs!  They should be caught and pinned to a collection board and labeled.  Check the State Fair Book to see what the reuirements are for collections.  

If your child is young and are interested in learning about entomology I recommend the curriculum books at Shop 4-H!  Elizabeth had a great time learning though this book.  You can enter the curriculum at the fair.  Take a look at the beginning of Diplay Exhibit Section of the 4-H State Fair Handbook for the requirements to enter them.  Last i checked it was 7 completed activities per year.  

Here is the link to some of the entomology and bee curriculum.

You can also enter anything having to do with bees, honey, or their wax.  One year the girls entered beeswax candles under this project area.