Thank you for sharing your concern for your grey cat who is vomiting, and for your other cats as well. It is clear you love them and want to see them well, and God sees that compassion. He cares for all of His creation, and He invites us to bring every burden to Him in prayer.
Scripture shows us the heart of Jesus for healing. When Jesus healed the sick and those tormented by disease, Matthew pointed to the prophecy of Isaiah that the Messiah would take our infirmities and bear our sicknesses. That prophecy is fulfilled in Jesus, and it shows that His suffering on the cross was not only for the forgiveness of sin but also for our physical brokenness. When we remember His body broken for us, we remember that healing, both spiritual and physical, comes from Him.
The Bible also teaches us to pray for the sick and to ask the Lord for mercy. James writes that if anyone among us is sick, they should call for the elders of the church to pray, anoint with oil in the name of the Lord, and the prayer offered in faith will help the sick and the Lord will raise them up. That promise is given for people, and it shows us that prayer for healing was a normal part of the early church and the ministry of Jesus and the apostles.
At the same time, Scripture is honest that God does not heal every sickness in this life in the way we ask. Paul prayed three times for the thorn in his flesh to be taken away, and the Lord did not remove it. Paul, who saw God work many miracles, also wrote that Epaphroditus was sick and nearly died before God had mercy on him, and that he had to leave Trophimus behind because he was ill. Even Paul spoke of his own infirmities. Godly people still experience sickness, and healing remains a sovereign work of God. His ways are beyond our understanding, and it is wrong to tell someone that sickness means they lack faith or have done something wrong. Not everyone is healed, and we do not always know why. Sometimes healing comes at once, sometimes it is a process, and sometimes the Lord in His wisdom allows the sickness to remain for a time.
We do know that God can heal and that He does heal, and every healing is a gift from His hand. We can trust His goodness even as we ask for His help.
So I will pray for you and for your cats. I pray that the Lord will have mercy on your grey cat, ease her vomiting, and restore her health if it is His will. I pray He will give you wisdom and guidance as you care for her, including wisdom to seek good veterinary care promptly if she continues to vomit, becomes lethargic, or cannot keep food and water down, since the Lord often works through practical means as well. I pray He will comfort you and give you peace as you wait on Him, and keep your other cats healthy and protected.
You can continue to bring this need to Jesus simply and honestly. Thank Him that He bore our sicknesses and that He is compassionate, ask Him for healing according to His will, and trust Him with the outcome, knowing that He hears and loves you.