Water Gun Race - Is it doable?

You haven't bought the stuff yet, so you still have choice of solenoids. It will probably be easier if you use 12vdc solenoid. You could even use the type used for drip irrigation you can get at any hardware shop. I hear clippard makes some nice small water valves.

You can drive a 12vdc solenoid pretty much the same as you drive a relay, low side switch NPN transistor and flyback transistor.

You mention using a reservoir. I take it this won't be connected to a water main system, but will be portable?

If it were me, I would probably not use a pump. I would probably use a pressurized beer keg. Probably would stay away from carbon dioxide and use nitrogen because I'm not sure how carbonated water acts when you shoot it. It might scatter and not form a nice stream. I'd probably use a nitrogen tank or bring my little air compressor. But if you don't have any experience with compressed air it's probably not the best option.

Remember, you only need a tiny narrow stream because the tube is filled by its own valve.

And don't forget about the tube emptying valve!