Do I need it for the Indianapolis race? I got the car via trade and it had every upgrade except this one, so I'm wondering if there's a particular reason the trader had left it out.
No, just max out the ZR1 with the RM, RS tires, etc and you will do great. I found (in GT4 at least) that rigidity improvement made the cars a little understeery after it was installed.
If you stiffen the chassis with rigidity improvements then understeer may be introduced as your handling set-up will give different results with a stiffer chassis, you may need to rejig your settings to allow for that extra stiffness (if GT5 is that realistic that is)