I must admit I choked up when I just read it.

First, here is the text of Rainbow Bridge:
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Just this side of heaven is a place called Rainbow Bridge. When an animal dies that has been especially close to someone here, your pet goes to Rainbow Bridge. There are meadows and hills for all of our special friends so they can run and play together. There is plenty of food, water, and sunshine, and friends are warm and comfortable. All the animals who have been ill and old are restored to health and strength, those who were hurt are made better and strong again, like we remember them before they go to heaven. They are happy and content except for one small thing—they each miss someone very special to them who had to be left behind. They all run and play together, but the day comes when one suddenly stops and looks into the distance. His bright eyes are shining, his body shakes. Suddenly he begins to run from the herd, rushing over the grass, his legs carrying him faster and faster, and when you and your special friend finally meet, you cuddle in a happy hug never to be apart again. You and your pet are in tears. Your hands again cuddle his head and you look again into his trusting eyes, so long gone from life, but never absent from your heart, and then you cross the Rainbow Bridge together.
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And here is the article about finding the original author:

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The ‘Rainbow Bridge’ has comforted millions of pet parents. Who wrote it?
More than a dozen people claimed ownership of the beloved poem, but the real author had no idea what a global sensation it became.
By Rachel Nuwer - Published 23 Feb 2023

If you’ve lost a pet, you have likely encountered “Rainbow Bridge”—a simple but poignant poem about animal heaven and the promise of reunion with furry loved ones. Copies of the poem are regularly given to bereft clients by veterinary hospitals; references commonly appear in condolence cards and social media messages to grieving pet parents.

For all the millions of lives “Rainbow Bridge” has touched, though, the author of the poem has remained unknown—until now. She is Edna Clyne-Rekhy, an 82-year-old Scottish artist and animal lover. Until recently, she had no idea that the poem she wrote over 60 years ago—to honour her dog, Major—had brought comfort to so many others.

“I’m absolutely stunned,” she says. “I’m still in a state of shock.”

Clyne-Rekhy’s authorship likely would have been lost to history were it not for the tenacious sleuthing of Paul Koudounaris, an art historian, author, and cat owner in Tucson, Arizona. Koudounaris has spent the past decade working on a book about pet cemeteries and frequently encountered references to the “Rainbow Bridge” in his research.

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https://www.nationalgeographic.co.uk/animals/2023/02/the-rainbow-bridge-has-comforted-millions-of-pet-parents-who-wrote-it