I come from an organized (SDA) church. We believe in immersion. The pastor usually does it as head of the church. Sometimes an Elder does it (under the jurisdiction of the pastor) for various reasons, one being our pastor has three churches, another it was his grandson.
But I would say that the father as the head of the home/church would be very blessed to be able to baptize his own children. My nephews were baptized in a river.
I think its wonderful that your children understand and have requested baptism.
"When Christ uttered the cry, 'It is finished,' he knew that the battle was won. As a moral conqueror, he planted his banner on the eternal heights. Was there not joy among the angels? Not a son nor a daughter of Adam but could now lay hold on the merits of the spotless son of God, and say: 'Christ has died for me. He is my saviour...'" - Ellen G. White.