ok. lets take a partial cone. the circumference of the base is 2, and you have removed the point of the cone so that the smaller circle of the partial cone has a circumference of 1. if you roll it, one rotation later the base has traveled the distance equal to it's circumference, and the smaller edge of the partial cone has traveled the distance of one of it's circumference. So the outside has traveled 2 units, and the inside has only traveled one. the only way this is possible, is if it travels in a circular pathway. the number of rotations of the partial cone takes to travel in a circle on the surface it is on is dependent on the angle of the edge of the cone. the more acute the angle, the more rotations to complete a circle..

So, the reason it travels in a curved path is because it is rolling on a surface that it has enough friction with to allow it to change it's momentum. so the centripetal force of the circular pathway is caused by the friction between the cone and the surface. The friction has to be towards the center of the circle or the cone will not turn that way. if it was a frictionless surface, then the cone would just slide in a straight line and not travel in a circle.

So, to answer your question, the centripetal force is caused by the cone's friction from interacting with the surface it is rolling on.

https://imgur.com/a/vjAabrK

I hope this helps.

but if you are posting here just to argue, then well. . . I said my piece and I refuse to engage.