The first step to Sainthood which is conducted by the Vatican is the status of Venerable. To attain this state of veneration, the prospective Saint must be declared to have lived a life of heroic virtue. The youngest non-martyr to have been declared thus was Jacinta Marto, one of the visionaries of Fatima. Many involved of the process actually saw her age as an impediment, as they were unsure if someone so young could be declared as having lived a life of heroic virtue without being martyred. The primary reason that the Venerable status was originally granted was due to the fact that she, along with the other visionaries, were threatened with martyrdom to recant of the Visions and yet they still stood firm in their faith. They readily accepted the potential martyrdom. Jacinta was 10 when she died, past the age of reason. Both the acceptance of martyrdom and the fact that she was past the age of reason allowed the commission to rule that she lived a life of heroic virtue, thus allowing for the subsequent Beatification and Canonization.
Infants, as they have neither not yet reached the age of reason nor could they to their own martyrdom, cannot fulfill the requirements for Venerable status. There would have to be a special indult by the Pope to allow the Venerable status to be granted or for it to be skipped in favor of direct initiation of the process for Beatification.
This is not to say that private veneration is not possible. There are multitudes of people in heaven who are not recognized by the Church. They may be a saint, but not a Saint, and still intercede for us on earth.