No, it's exerted in this case by the wall behind you keeping you from flying away. If you were on a string attached to the center, it would be the string exerting the centripetal force. You keep going off on tangential paths, but the wall is there and shoves you back toward the central point. every instant. Inertia and friction and gravity are what keep you in place on the wall as the machine accelerates.
I think that the force exerted by the wall on the rider is a response to the force exerted by the rider on the wall, and not vice versa. The wall's centripetal force is opposed by the walls and ceiling of the ride. If this was not the case, then the ride would collapse if no riders were present.
Sometimes I wish I didn't know now the things I didn't know then.
