Circular motion of a particle around a track -- what provides the centripital acceleration?

[A.T.]

If there was no friction then the object would fly off the circular trajectory and continue in the direction of the tangential velocity.

Yes. the car would move tangentially. But the relative motion between road and the bottom of the rotating wheel would radial (red arrow below). If present, static friction opposes that motion, and acts radially, thus providing a centripetal force to the car.

 

[Physics guy]

As I turn the wheels to the left the wheels are pushing to the right against the floor. Friction allows the floor to push back against the wheels allowing them to turn left. If there was no friction then the object would fly off the circular trajectory and continue in the direction of the tangential velocity.

Yes you got it right

 

[Lnewqban]

... If there was no friction then the object center of mass of the object would fly off the circular trajectory and continue in the direction of the tangential velocity.

... the center of mass of the object.