Can a person earn a private pilot certificate in a light-sport aircraft exclusively, or must some of the training be in a more "conventional" aircraft?

A light-sport aircraft can be used for training of all pilot levels-provided the aircraft is properly equipped for the training required and the LSA manufacturer does not specifically prohibit such training in that aircraft. Special LSA aircraft fall under direct control of the manufacturer, so even if the FAA might allow a particular operation, it must also be approved by the manufacturer in order for the aircraft to be operated in that capacity.

Private Pilot Training in an LSA? | Flying Magazine