Seeing as how in both the original anime and the manga/brotherhood, the power in them came from human souls, I'd venture to guess that wouldn't work.

What it comes down to isn't just the soul, but knowledge itself. Knowledge is the ultimate power of the series. Or, so it seems to make argument for. We see essentially all throughout the story that knowledge and science-like discovery brings great wonders and horrible terrors. Animals don't hold much information or memory. It's not that they couldn't use animals for philosopher stones. It's more likely an efficiency problem, or like you pointed out they'd just use animals.

Another reason why human transmutation is prioritized is because it's a lot more taboo. You're not gonna feel the same stakes watching a show where some alchemists make a philosopher's burger out of cows and use it to get their bodies back. Tada! story over. Humans don't care much for animal life the same way. We already benefit from the mass slaughter of animals. It's when humans see themselves in the slaughterhouse that horror is properly introduced. It compels the viewer and the main characters to find more ethical alternatives (eventually discovering that with alchemy, there are no ethical alternatives).

That's why the ending is about sacrificing knowledge to regain a human body and soul. It brings the conversation full circle.

I think they don't have souls in the show or something

That warrants a plain old "why". What would be the advantage you see in that plot twist?

And why do I think that would be pointless? I only know the Brotherhood incarnation but it is made very clear that Father tried to make itself perfect by expunging from his soul the Seven Deadly Sins, which are traits or tendencies only humans can develop. Sins are by default moral options underlined by a notion of good/bad, something other animal do not have. Animals understand something as good or bad from the consequences and outcomes a given action returns.

By contrast, humans have the notion of good/bad as our much more complex brain can project and predict consequences from a given option. This makes the humans eligible as Philosopher's Stone ingredients because of thar extra sophistication.

And while writing this I remember that ar some points Roy Mustang talks with the souls trapped in the Philosopher's Stone inside him to know if the souls are willing to allow him to use their power. The human conscience is apparently still there, which might also be a clue for why humans are material for stones and animals are not.