Indeed, animism would be one very coherent way to approach it.
It's my preferred approach because it allows me to have supernatural elements without treating them as supernatural. Explanation:
I write both fantasy set in a created world and horror/dark fantasy set in the real world with fantastical bits added. In both sets of stories, my MCs are beings native to the weirdness, who know how it works, and who don't find it odd at all.
In the horror stories, the dead walk, ghosts haunt, vampires and werecritters do their thing, witches and occultists cast spells and summon demons, all just because they do. The arcane secret of this fictional reality is that all such things are real and that's how the universe works. This world is founded on the premise that souls are real, immortal and everywhere, and life and death are just transitions between temporary manifestations in various forms. Evil and horror arise not from the reality of, say, walking corpses, but from evil intentions -- malice, hate, greed, and so forth. There is a balance of energies to be maintained for the good of all beings, and evil people tend to seek to upset that balance, usually because they are short-sighted in their greed or ambition. This results in bad stuff which my characters have to fix.
It's a basically animistic world view, of the dualistic Indo-European type. A similar example might be the Nightwatch trilogy, in which good spirits/witches/psychics are balanced against evil spirits/witches/psychics, forever policing each other to keep the balance that maintains the safety of life.
No deity is required for this kind of supernatural universe because it presumes this is simply the nature of reality. Most people don't know it because most people don't have the ability to mentally tune into the energy frequencies that allow them to see the spiritual plane or dimension that exists all around us. But just because people are ignorant of the nature of their universe, that doesn't make that nature really supernatural or magical or dependent on religion.
In my fantasy stories, I envisioned a world in which magic is the result of manipulating that fictional "fifth force." I created a type of energy/radiation/whatever -- MacGuffin waves -- that is a feature of the natural world, and in this fantasy reality beings evolved to be able to sense and manipulate that energy. Even plants and animals do it, and sentient beings do it a lot. The culture of this world is entirely founded on magic, which functions in their modern world as electricity and the internal combustion engine function in the real modern world.
This fantasy world does have gods, but their function is completely different from the concept of a cosmic, omnipotent creator deity. The gods of this world are really just extremely advanced mystics who channel the magical energies of the world through a consciousness prism, in which the ways that the energy manifests in the phenomenological world attain conscious awareness and direction.
For example, the magical elements of air, fire, water, and spirit coalesce into consciousness and form the continents (or reform them after a massive geological upheaval; this is not a creation myth). That elemental consciousness, which implies that all sentience is of similar nature, remains intact through the bodies of psychic mystics, who become the oracles of what are called gods. And the oracles are also called gods. We can think of them as just super-rarified wizards. We see somewhat similar concepts in Hinduism and some Buddhist traditions, with living people seen as avatars of some deity or spiritual principle.
In my fantasy world, the gods are not worshipped per se. They are consulted for guidance in many cases. In some cases they are placated to keep the energies peaceful. In all cases, they are specialized manifestations of the energy field, so they are in charge of whatever they are the god of. Some are the guardians of their continents. Some are in charge of plant and animal life, or the weather, or the stuff that people do such as law and government, healing, arts, sciences, commerce, etc.
So one my characters, needing a storm to break out at a given moment, would not pray for a deity to create a storm for him. He'd cast a spell or use an enchanted object to create his storm himself. But if he screwed with the weather so much that the god that manages weather got pissed off at him, said god would curse him, i.e. cause his energies to become so disarranged that he suffers for it. He would have to do something to placate that god.
It's an animistic system that assumes that spiritual power is natural and available to all beings, that everything has someone or something in charge of it that we have to deal with or work around, and that essentially tends to view gods as similar to other kinds of beings except with more power and/or more responsibility -- but
not in charge of the destinies of other beings.