That's a drive mode. A continuous exposure sounds more like bulb mode.
Anyway, you're not SOL, but life is made much easier with a more powerful flash and a camera that lets you go up to those purdy high ISOs. I can easily shoot with high-speed continuous drive on my 7D with a 580EXII. You should note, however, that if you use ETTL or iTTL, it will slow-down the camera because the it needs time to calculate the flash exposure, as well as fire the pre-flash.
What will happen is that if you fire the flash a whole lot, it will run-down the capacitors. If the flash is fired at around 1/8 or 1/16, this won't be a practical concern (at 1/8 power and ISO 1000, a 580EXII reports that proper exposure falls around 13m, hence why high ISOs are so useful for high-speed flash shooting). The flash won't however, be totally useless, as Plato puts it.
To further correct the inaccuracies above, only the recycle time of the flash at 1/1 power would be too slow to fire successive frames. Bring it below 1/2 and the capacitors will draw power from the batteries while providing power to the flash bulb for each firing, in essence creating a buffer between the higher-resistance batteries and the flash bulb. Eventually, the capacitors will run out of sufficient power to fire the flash, but as I said, once you get to around 1/16, this really isn't a practical concern at all, and flash units like the 550EX, 580EXII, SB-800, and SB-900 can all provide quite a bit of power at 1/16, as long as you boost the ISO and open your aperture sufficiently.
However, since you're creating a stroboscopic effect with your flash unit, in all likelihood anyone you're shooting will not appreciate what you're doing. If you must use flash at a sporting event, and are allowed to do so by the organizers, it would be better to work on your timing.