Thank you for sharing this burden and for your care for your friend and her beloved dog. It is a tender thing to love an animal that has become part of the family, and it is right to bring that concern to the Lord in prayer.
Scripture teaches that true wisdom does not come from human learning alone but from God himself. The fear of the Lord is the beginning of wisdom, and wisdom is more valuable than riches or rubies. Knowledge can give us facts, but wisdom gives understanding for how to use what we know. In Christ are hidden all the treasures of wisdom and knowledge, and God invites anyone who lacks wisdom to ask Him, and He gives generously. That is what Solomon did when he asked for a wise and understanding heart to care for those entrusted to him, and God was pleased to answer. The wisdom that comes from above is pure and peaceable and leads to what is good, very different from the wisdom of this world which comes to nothing.
We also see in Scripture that prayer for the sick was part of the life of Jesus and the early church. They brought those who were sick to Jesus, and later the apostles prayed for the sick, and James instructs those who are sick to call for the elders to pray and anoint with oil in the name of the Lord, with faith and trust in God. We do not always understand why healing comes in some cases and not in others, but we do know that God hears prayer and that we can trust His goodness and power.
So I am praying with you in the name of Jesus for this dear dog, asking the Lord to show mercy and to bring healing according to His will. I am also praying that God, who is the source of all true wisdom and understanding, will grant clear insight and discernment to the veterinarian, to understand what is wrong and to know the best way to help. May the Lord give skill to the hands that care for her and peace and comfort to your friend who loves her.
Father, in the name of Jesus we lift up this sick dog to You. You know how much she is loved and You care for all that You have made. We ask for Your healing touch upon her and for wisdom from above for the doctor who is seeking to help her, that You would guide his mind and decisions. We trust her into Your hands and ask for Your peace for all who are worried for her. Amen.