I'm going to say no. You can play single player, but once you hit the ladder and multiplayer, the mouse control, speed and accuracy will greatly effectaffect what you can do and how you can compete at certain levels.
Short answer: Your micro will be weak and your macro could technically take a hit as well.
Advantage: Playing with a touchpad could potentially help with your hotkeys and you will be in a good position when you do get a mouse.