"Who knows if the spirit of man rises upward and if the spirit of the animal goes down into the earth?'" (Ecclesiastes 3:21)
It's sort of ambiguous,
There are some words used in relation to animals that can mean 'soul',
ne·fesh = a soul, living being, life, self, person, desire, passion, appetite, emotion
Various translations concerning animals translate it as 'has life' or 'the breath of life in it' or 'in which is a living soul' or 'in which is a living spirit'.
They are described as living, translated:
chai·yah = alive, living
They are described as having a spirit
ru·ach = breath, wind, spirit
The spirit is what makes one's body alive,
"As the body without the spirit is dead, so faith without deeds is dead." (James 2:26)
"How many are your works, O Lord!
In wisdom you made them all;
the earth is full of your creatures.
There is the sea, vast and spacious,
teeming with creatures beyond number—
living things both large and small.
There the ships go to and fro,
and the leviathan, which you formed to frolic there.
These all look to you
to give them their food at the proper time.
When you give it to them,
they gather it up;
when you open your hand,
they are satisfied with good things.
When you hide your face,
they are terrified;
when you take away their breath,
they die and return to the dust.
When you send your Spirit,
they are created,
and you renew the face of the earth." (Psalm 104:24-30)
But who knows if it's their body that returns to dust, and their spirit returns to God.
"In whose hand is the soul of every living thing, and the breath of all mankind." (Job 12:10 KJV)
There's are horses in heaven,
"I saw heaven standing open and there before me was a white horse, whose rider is called Faithful and True. With justice he judges and makes war. His eyes are like blazing fire, and on his head are many crowns. He has a name written on him that no one knows but he himself. He is dressed in a robe dipped in blood, and his name is the Word of God. The armies of heaven were following him, riding on white horses and dressed in fine linen, white and clean." (Revelation 19:11-14)
"This is the gospel that you heard and that has been proclaimed to every creature under heaven, and of which I, Paul, have become a servant." (Colossians 1:23)
"He said to them, "Go into all the world and preach the good news to all creation." (Mark 16:15)
That's why St.Francis famously preached to sparrows.
"For the creation was subjected to frustration, not by its own choice, but by the will of the one who subjected it, in hope that the creation itself will be liberated from its bondage to decay and brought into the glorious freedom of the children of God. We know that the whole creation has been groaning as in the pains of childbirth right up to the present time." (Romans 8:20-22)
"Then I saw a new heaven and a new earth, for the first heaven and the first earth had passed away,..."(Revelation 21:1)
Surely this includes animals?
"that at the name of Jesus every knee should bow,
in heaven and on earth and under the earth,
and every tongue confess that Jesus Christ is Lord,
to the glory of God the Father." (Philippians 2:10-11)
"Then I heard every creature in heaven and on earth and under the earth and on the sea, and all that is in them, singing: "To him who sits on the throne and to the Lamb be praise and honor and glory and power, for ever and ever!'" (Revelation 5:13)
"My mouth will speak in praise of the LORD. Let every creature praise his holy name for ever and ever." (Psalm 145:21)
How could they do that unless they're included in heaven and the new earth?
Psalm 104 gives animals sentience. On animal sentience in science, elephants have been found to comtemplate themselves in mirrors,
Elephant Self-Awareness Mirrors Humans | LiveScience