Yes. Shimano 9 speed drivetrain stuff is interchangable, even between mountain and road. You won't need a new rear derailleur* even though Sora is technically 8 speed, as all the gear indexing is done by the shifter.
* the only time you'd need a new one is if you are going to a much wider ratio cassette, or from a double crank to a triple crank. In those cases you need a longer-cage derailleur to take up the extra chain slack that can happen in the [now more] extreme [than before] differences in gear teeth.
A note on braking systems though: V-brakes and road bike caliper brakes require different cable leverage. The mountain bike STI levers are set up for V-brake cable leverage, and the road bike ones set up for road bike style dual-pivot caliper brake leverage.
As you have a flat-bar road bike with V-brakes, this most likely won't be a problem: you could just get any old mountain bike V-brake levers to use with what you have. (I'm guessing your current brake levers and shifters are one integrated unit)