Term for belief in the existence of souls
Solution 1:
Within philosophy, particularly the philosophy of the mind it's called "dualism" because it's the idea that we are made up of two parts, a material body and an immaterial mind ("soul").
Dualists are contrasted with "materialists" and "idealists" which both have very different meanings in philosophy jargon from their conventional usage. Materialists think the mind is a result of material processes while idealists think that minds are all that exist (or at least all that we can know to exist) and the material world may well be an illusion. All of this is roughly speaking of course. Being philosophy, there is more that can be said about it, and much more still which is said about it.
The term "dualism" would not be readily understood that way outside of this context though.
Solution 2:
I would term this “Animism” which has three meanings, all related to a common etymology:
OED
animism, n.
Etymology: < classical Latin anima life, soul
- Philosophy. Any of various theories postulating that an animating principle, as distinct from physical processes (chemical, mechanical, etc.), directs the energy that moves living beings and governs their growth and evolution; = vitalism n. An influential version was propounded by the German chemist and physician Georg Ernst Stahl (1660–1734) and enjoyed a short-lived revival through the writings of the French philosopher Francisque Bouillier (1813–99).
1871 E. B. Tylor Primitive Culture I. xi. 384 I purpose here, under the name of Animism, to investigate the deep-lying doctrine of Spiritual Beings, which embodies the very essence of Spiritualistic as opposed to Materialistic philosophy... The sense of Spiritualism in..the general doctrine of spiritual beings, is here given to Animism.
2004 tr. A. Lalande in N. Gross & R. A. Jones Durkheim's Philos. Lect. lxxii. 286 More recently,..animism has been revived with more moderation and good sense by Francisque Bouiller in a book titled The Vital Principle and the Thinking Soul.
2. The attribution of life and personality (and sometimes a soul) to inanimate objects and natural phenomena; = animatism n.
1866 Fortn. Rev. 15 Aug. 84 The theory which endows the phenomena of nature with personal life might be conveniently called Animism.
1871 E. B. Tylor Primitive Culture I. 45 The animism of the ruder tribes of India.
1973 Jrnl. Genetic Psychol. 123 219 Piaget..observed that many preadolescent children tend to ascribe life and consciousness to inert objects, a phenomenon that he called ‘animism’.
1999 Piecework Nov.–Dec. 23 Their folk religion was based on animism or the attribution of a spirit or conscious life to material forms.
3. Spiritualism and Philosophy. Belief in the existence of a spiritual world, and of soul or spirit apart from matter; spiritualism as opposed to materialism.
1869 Appleton's Jrnl. 31 July 567/1 Animism is the doctrine of all men who believe in active spiritual beings; it is essentially the antagonist of materialism.
1880 J. Rae in Contemp. Rev. Oct. 615 The universality of what Mr. Tylor calls Animism, the belief in spiritual and unseen agencies.
Solution 3:
The one true word for "soul" in Greek is psukhê, and so correct English formations would be psychism and apsychism. It is true that psukhê could have many meanings, primarily "life, ghost, soul, the conscious and feeling self", etc. But it is the most obvious translation, and the only one I can think of that has spiritual connotations.