The nearest place where you can find animals in Riverside is almost outside the town, at a local farm.
In this barn, as shown in the photo, I found 2 cows.
On this farm, you’ll also find animals.

I found 3 places in West Point with animals, they were also in farms far from the area.

Just ahead, there's another chicken coop...

Just a bit further ahead, there's another barn.
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I didn’t fully explore the town on my map, but it’s just below the farm.

Just above the trailers, you'll find a farm.

Heading a bit further up the main road from Rosewood, you can find this chicken coop too.

Maybe you’ll find this abandoned. Just press ‘V’ to see if the animal icon appears. You can usually tow them with the animals inside, but they get angry when they get out and you’ll probably get kicked. But if you find a trailer with an animal inside, you can remove them from there with no problem at all.