The British singer, 41, posted an Instagram video showing a man appearing to remove suitcases from the vehicle on Friday at 05:42 local time.
Mel B asked fans to contact her husband Stephen Belafonte via Instagram if they had any information about the incident.
She said police were investigating but the couple wanted the public's help.
"The police already have a good idea who you are, but in the meantime anybody who wants to make that 5000 dollars, DM my hubby right now," said the singer, who has been hosting The X Factor in Australia.
Mel B said the couple would wire the money across to anyone who helped them.
The singer, real name Melanie Brown, is reuniting with fellow former Spice Girls Geri Horner and Emma Bunton in a group known as GEM.