Let's see...20W at 9V is about 2.2A. The LM7809 is usually specified for up to 1A, and has built in current limiting.
I think the regulator would survive the attempt (as in it won't be destroyed), but by shutting itself down. The 780x regulators are pretty hard to kill.
Your heat sinking needs to be able to dump about 7W of power (3ish V * 2.2A). The junction-to-case thermal resistance of the 780x is listed as 5°C / W, so the junction would be about 35°C warmer than the heat sink assuming a very good thermal bond between it and the case. This means that the heat sink must not get warmer than 90°C to keep the device within its maximum rating of 125°C. (These details may vary from manufacturer to manufacturer, in fact; there isn't just one 7809 design.)
One 7809 per panel might work, though it's on the edge of the chips specifications, and sharing a heat sink may be feasible. You still have to remove the same total amount of heat, of course, as the power dissipated by the regulators remains the same.
Another option may be to use a PWM circuit to reduce the average power consumption of the LEDs to something reasonable. You could probably do this with a 555 and a power transistor and a few minor interface bits. That would be more efficient (less heat), though would probably need some sort of filtering to prevent excess RFI.
