It's a curious question, whether Buddhism is a religion or not, or whether it's a spiritual tradition. Religion and spirituality deal with God and the Soul. These are rather elusive in Buddhism. Buddhism does cultivate the realization of the ultimate nature of the universe and the self. But it's our habitual grasping at fixed concepts of these ultimates that is the root of our misery. Faith in Buddhism is the faith to let go.
Modern science is devoted to the quest for the ultimate nature of the universe, while it has discarded as meaningless the quest for the ultimate nature of the self. Given the overwhelmingly powerful role of science and technology in today's world, many who still find meaning in the self look for suitable roles for such an entity within the framework of science. The collapse of the wave function provides one popular such role.
While quantum mechanics has given scientists remarkable powerful methods to understand phenomena at the scale of molecules, atoms, and nuclei, it raises many difficult questions about the nature of the measurement process, and the nature of the phenomena being measured. When not being measured, physical systems seem to be able to explore many trajectories simultaneously through their spaces of possible configurations. But a measurement will trap the system in just one configuration, from which it will then evolve further after the measurement.
The fundamental paradox is that any measurement apparatus is itself just another physical system. The physical system being measured and the physical system performing the measurement, together form just a single larger physical system. Quantum mechanically, this larger system is just as capable of exploring multiple trajectories through its richer space of configurations. The collapse of the wave function, the pruning of possibilities down to a single actuality, is not apparently a consequence of the fundamental equations by which quantum mechanics describes the evolution of phenomena. The collapse seems to require some outside agent - which seems like a perfect role for a soul or spirit or self or consciousness.
Doubt and Certainty, by Tony Rothman and George Sudarshan, discusses various examples of the asymmetry of the direction of time - i.e. the world looks quite different when a movie is run backwards. One classic asymmetry is thermodynamic, e.g. a heat flows from a hot cup of tea out into the surrounding room, until the tea and the room reach the same temperature. It never happens that a cool cup of tea in a room gradually absorbs heat from the room until the tea is much hotter than the room. Another key asymmetry is the collapse of the wave function. There is no way to unmeasure a system! Rothman and Sudarshan suggest that maybe these two asymmetries are really just one, that the collapse of the wave function is actually some kind of thermodynamic affair. The pruning of possibilities might be more like an entropic scrambling. It could be a bit like erasing a chalk board. The chalk that formed the letters is still there, but the letters disappear because the chalk is so uniformly spread around.
Where does this entropic scrambling happen? Could it be buried in some deep perceptual mechanism, some homuncular observer at the foundation of the mind? Christine Skarda has proposed an intriguing perspective on perception, where the function of perception is to chop up what is originally interconnected into discrete objects. She also proposes that much of the chopping happens right at the surface, at the sense organs themselves, rather than at any fundamental homuncular level.
The collapse of the wave function will generally occur well before even the sense organs get involved - at whatever point the isolated microsystem interacts with coarse macroscopic systems like cameras, transistors, etc. One can imagine an experimental arrangement though where that first thermodynamic interaction is with the retina of the observer's eye, with the sense organ.