I am trying to make a line follower with the help of this tutorial. In the tutorial they have used Arduino motor shield. I want to know, is there any way to control the motors without using Ardumoto?
You can't connect a motor directly to an Arduino. The Arduino cannot supply enough current. You need a motor driver. But any motor driver circuit will do, it need not be that exact one. Be sure to get a motor driver with good documentation, so that you'll know how to write the code (sketch) to make it work.
Ardumotor is a very basic breakout of a standard 2 motor h bridge ic. Any H-bridge can replace it.