Besides Hawkgirl, all the other founding members of the Justice League in the DCAU had a musical theme. Plenty of other DCAU characters had them too (Aquaman, Green Arrow, Supergirl, Batgirl, almost every Batman villain).
There was never any theme I connected with Hawkgirl though. This might just be because I never managed to separate it out and associate it with her. I didn't really link the Martian Manhunter or Wonder Woman themes to their characters until it was pointed out to me.
I've tried looking on You Tube and there are plenty of compilations of DCAU music, but none of them seem to have a Hawkgirl theme.