How is existentialism portrayed in this book?

Through the narrative, we can understand that the theme of existentialism takes an ironic turn as Orwell progresses in the story further. The way animals living at the farm were initially controlled by the human farm owner, their inner need to express and exercise their freedom increased day by day. Therefore a major glimpse of existentialism can be witnessed at that point when the spark of rebellion ignited among the animals to overthrow human control, and getting a free reign on their farm. As the true essence of the existential autonomy is that people govern their lives by making their own choices and decisions, animals thwarted the totalitarian concept in favor of the free-willed, liberal approach. However, the same freedom movement came to bite its followers when a selected few from their own clan took control over the farm as quoted in the text:
“This work was strictly voluntary, but any animal who absented himself from it would have his rations reduced by half.” ― George Orwell, Animal Farm
Thus the existentialism took an ironic turn when it got exercised selectively for the pig leadership only.

While existentialism isn't an overt theme in the book, there are elements of it, nonetheless. One such element is the need to impose meaning on an inherently meaningless world. That's precisely what the animals do when they take over the running of the farm from Mr. Jones. Where once they were slaves, now the animals have meaning and purpose in their lives.
Existential freedom is another important element. In bringing about the Animalist revolution, the animals are making a conscious choice to take control of their destiny, affirming their existential freedom to make their own decisions in life instead of running away from it as, according to existentialist thinkers, so many of us often do.
Having made this momentous decision, the animals must then live with the consequences of their actions. This helps to explain why so many of them are reluctant to challenge Napoleon's regime, even when things start to go wrong. All the animals have taken ownership of the revolution; it's an expression of the existential identity they've constructed for themselves. They've come to feel that this is their true identity, true because it has been freely chosen, and that in existentialist terms, it would be inauthentic of them to walk away from what they truly are.