They can power the OI with a 9V battery if they want a portable OI for demonstrations, but they must realized that during competition, all connections to the robot go through the Competition port, including the power to use the OI. So the answer the question:
In Competition: No
At School: Yes, but notice that any voltage changes will cause your analog sensors to mess up on your OI like your joysticks, so be careful, and I highly advice you to make a voltage regulator that will ensure a steady voltage.
One more thing to add: If you’re using something like the USB Chicklet (http://www.ifirobotics.com/usb-chicklet.shtml) on your operators console, there is no place to plug it in, and you MUST bring a battery for it to the competition. the OI gets it’s power from the “competition” port, but no other independently powered devices will!
Voltage regulators have a parameter called “Dropout”- the input voltage at which regulation no longer occurs.
A 78xx series regulator requires an “overhead” - the voltage difference between input and output - of something like 2 or 3 volts to avoid dropout. Low-Dropout (LDO) regulators exist, with overheads of a fraction of a volt.
Your 9 volt transistor battery was very close to the dropout limit for a 7805 regulator already. If you draw more current from the battery tha its rating - I think it’s 20 mA for that kind of battery - the battery output voltage can sag (get lower). You may want to check what current the OI draws, either by looking at the specs or by direct measurement, and decide if a two 9 volt batteries in parallel is sufficient to power the OI reliably.