The original iDVD program was an OS 9 application. That is why you see a copy of it in "Applications (OS 9)". But because of the way it works, it only runs when booted directly into OS 9, it does not work in OS X's "Classic" mode.
If iDVD came on your G4, then your G4 has a SuperDrive, it should be able to read and write DVDs. If it can't, then your drive is broken.
But, just to make sure (for example, if you're not the original owner, it's possible that the previous owner removed the SuperDrive and put a CD-ROM or CD-RW drive in; or it may not have had a SuperDrive in the first place, and someone installed the software from a system that did.)
The way to find out what is in your computer:

menu, "About This Mac". Click "More Info..." (This loads System Profiler.) In System Profiler, click "Disc Burner". Among other things, it will have a line for "Reads DVD" that will say yes or no, and a line for "CD-Write" and "DVD-Write" that show which types of discs it can write.
For example, my old iMac G3 says:
Reads DVD: No
CD-Write: -R, -RW
While the external drive attached to my MacBook Pro says:
Reads DVD: Yes
CD-Write: -R, -RW
DVD-Write: -R, -RAM, -RW, +R, +RW, +R DL
If your computer says "Reads DVD: Yes", yet it won't, then the drive is broken.