Are there any brands of external DVDRW drives, other than Lacie, that will work with my Mac G4 desktop and OS10.3.9?

Most will. There is a drive compatibility database at www.xlr8yourmac.com where you can read user reports with different drives and different Mac OS versions. There's a generic driver available that enables Macs to recognize nearly all drives, although I forget its name. You'll see it referred to at xlr8yourmac.com.

LaCie has the advantage of providing firmware updates that can be flashed using a Mac whereas most others require connecting the drive to a PC for flashing firmware.