Can I claim an instrument rental for education credit in Illinois?

Yes, it's specifically covered in the rules.  Note that the purchase of an instrument would not count. 

"What are qualified education expenses?
In general, qualified education expenses are the amounts paid
in excess of $250 for tuition, book fees, and lab fees at the
school where the student is enrolled in kindergarten through
twelfth grade during the regular school year.
Specifically, qualified education expenses include the amount of
tuition (including summer school classes required for
elementary or secondary graduation requirements).
book fees paid for the rental of books that were required as a
part of the school’s education program.

lab fees paid for the use of supplies, equipment, materials, or
instruments that were required as part of a lab course in the
school’s education program.
For example, if you rented a musical instrument from the
school (not from a business)
for a class, or
for participation in an extracurricular activity that resulted
in a credit toward completion of the school’s education
program,then this rental expense qualifies as an education expense."

"What education expenses do not
qualify?
Education expenses that do not qualify for an education
expense credit include expenses paid
to a daycare, preschool, college, university, independent
tutoring service, or trade school.
for the purchase of supplies, books or equipment that
are not significantly used up during the school year (e.g.,
purchasing musical instruments, costumes for a play)"

https://www2.illinois.gov/rev/research/publications/pubs/Documents/pub-132.pdf