Some churches have traditions where people bring their pets or farm animals to the priest or minister for a ritual blessing.  You will not find this tradition practiced in most Protestant churches.  Why not?

First, we do not find any clear example of such a practice in the Bible.  God prescribed many elaborate rituals for worship in Israel in the Old Testament, but all those traditions and rituals deal with our personal relationship with God.  The only reference to animals in those rituals were the animals offered in sacrifice for human sin.  Also, there is nothing in the New Testament which suggests that Christ or the Apostles offered blessings for domesticated animals.

Second, rituals practiced in church worship tend to evolve into sacraments.  Christ prescribed only two sacraments, Baptism and the Lord's Supper, which communicate God's forgiveness of our sins (see Matthew 26.26-28 and Matthew 28.19).   However, churches which emphasize tradition over the Bible list six or seven major sacraments (including confirmation, ordination, and last-rites) and an unspecified number of minor sacraments.  This practice raises human rituals to the status of divine ordinances through which God bestows His grace.  This we dare not do.

Third, when we practice rituals of pet-blessing, we imply that pets have spiritual qualities which the Bible says God gave only to angels and to people.  While we may see some human-like qualities in our critters, we dare not teach more than God has revealed about this.

We love our pets, and when they experience distress, we naturally ask God to intervene on their behalf.  The Bible says that God cares for wild animals (see Matthew 6.26) and He promises fertility to the faithful farmer's livestock (Deuteronomy 28.11).  These blessings come from God without our prayers.  Rather, the Bible teaches us to pray for God to help us be faithful to Him.  Animals don't have a problem with trusting and obeying God; we people do!