How does a mobile phone vibrate without any external force?

That's not true, Newtons's laws do not say that. What's important here is conservation of momentum. Inside the phone, there is an oscillating mass. While the mass inside has a momentum and thus a velocity in one direction, the (friction-free) phone has to have the same momentum in the opposite direction. It "vibrates".

Homework: Get on a skateboard (best kneeling, not standing), take a decent mass with you (e.g. a cobblestone) and move it forth and back in front of your chest. Now, put a large cardboard box over your head (e.g. from a refrigerator) and you have a box that moves back and forth without any external force.

If you want translation instead of oscillation, you have to divide the object, making one part go in one direction and the other in the opposite direction (again, with the same momentum). That's how rockets work, by expelling the reaction products of their fuel at high speed in the opposite direction. Again, without "external" force.

Alternatively, you can just sit in a chair, and punch the air really fast. When your arm moves out, your body moves back, when your arm moves back in, your body moves toward the arm.

At least one mobile phone I've heard about uses an unbalanced spinning weight. As the weight moves in one direction, the phone moves in the other, in accordance with Newton's Third Law of Motion.