Although crossing science and spirituality at anytime is usually something that is destined for dissonance and the burden of interpretation, by laws of what is known of the soul on one side and science on the other, then yes - reincarnation is scientifically possible.
It is a well known fact throughout the expanse of the universe that nothing is wasted. Everything has a purpose and everything is recyclable. To outright say something is impossible is simply asking someone to provide a proper explanation to what was previously unknown. From a flat planet, to a light speed limit, to many other things, all have proven to be something else. To say that the precious existence of a consciousness is also a one time occurrence, that seems like an unimaginable waste. In a universe that wastes nothing, how can the drive of life, the highest form of physical representation yet seen, be something uniquely trivial?
Ignoring the soul for a moment, there is energy in all living beings. Energy is known to flow, and energy cannot be destroyed. Those two scientific facts surrounding energy are well known. If life and the imprint of physical being is reflected on the energy of life, then that imprint is capable of storing the reflection and carrying it just as easily as photons can carry information from one end of a fiber optic strand to another. Unrestricted by a single flow, isn't it possible for stored information within bio-energy to flow to another receptacle?
Trying to prove that the energy within a person is their soul is irrelevant. Life energy versus soul, it's the same thing in principle. Some scientists have already determined that when a person dies, they lose a little weight from their bodies. Whether that weight is due to air leaving the body, or the energy dissipating, it hasn't been confirmed as yet, but it is something to think about.
Another thing to think about is the scientific possibility of teleportation. Teleportation has already been proven a reality and scientists in specialized facilities have been learning the art of teleporting electrons and other small particles with no apparent progress towards something of public application. However, if energy can be teleported - rather than simply flowing - then it is conceivable for the soul to teleport to another suitable vessel, just before that new form of life creates its own. The emergence of old information in a different location is just one of these facts that confirm the theory. Yes, reincarnation is scientifically possible.