Saint Augustine testifies that baptized infants received Holy Communion in the early Church. This only makes sense. If they’re regenerate, why can’t they receive the Christ in the Blessed Sacrament? Baptized infants continue to receive the Eucharist in the Eastern Catholic Churches.
Alas, I wish that my children could receive the Eucharist. My three year old and my five year old twins have often expressed their desire to "receive Jesus". They already have a Eucharistic faith and believe that the Eucharist is Jesus. Yet they cannot receive because they aren't "old enough". Saint Pius X, pray for us.
Here's the Augustine quote:
“Those who say that infancy has nothing in it for Jesus to save, are denying that Christ is Jesus for all believing infants. Those, I repeat, who say that infancy has nothing in it for Jesus to save, are saying nothing else than that for believing infants, infants that is who have been baptized in Christ, Christ the Lord is not Jesus. After all, what is Jesus? Jesus means Savior. Jesus is the Savior. Those whom he doesn’t save, having nothing to save in them, well for them he isn’t Jesus. Well now, if you can tolerate the idea that Christ is not Jesus for some persons who have been baptized, then I’m not sure your faith can be recognized as according with the sound rule. Yes, they’re infants, but they are his members. They’re infants, but they receive his sacraments. They are infants, but they share in his table, in order to have life in themselves.”Jonathon Deane has a great article on the practice of "paedocommunion" over at Called Communion.
Augustine, Sermon 174, 7