Animals have a material soul that ceases to exist when they die. Humans have a spiritual soul that continues to exist after death. But does this mean that animals will not be present in the afterlife? That is a question to which we do not have an answer. There does however seem to be hints that God will restore the universe at the end of time:
The visible universe, then, is itself destined to be transformed, "so that the world itself, restored to its original state, facing no further obstacles, should be at the service of the just," sharing their glorification in the risen Jesus Christ (CCC 1047).
As animals are part of the "visible universe," it seems possible that they too might share "their glorification in the risen Jesus Christ." It might be possible that God might also re-create those animals who have been a pleasure and comfort to man in his earthly journey. We don't know in this life, but it is something for which we can hope.
Explain to your children that heaven will have all that is needed for our perfect happiness. If they need their pet with them in heaven in order to be perfectly happy, their pet will be there. As adults we realize that God alone suffices for perfect happiness; but we also know that God, in his great love for us, gives us more than our mere necessities and that his love is reflected to us in his creation. It may be that our animal friends' presence in the next life will be God's gift to us.