Does Judaism present any rational proof for its belief in the existence of the soul?
The term "rational proof" is a bit problematic. If you mean conclusive proof, the answer is no. But this should not be surprising; there are many disciplines in which assumptions are operative even though they are not conclusive, all the more so regarding the non-physical realm. On the other hand, lack of conclusive proof doesn't render the idea irrational – there may still be a rational basis for accepting the belief.
The Kuzari, by Rabbi Yehuda HaLevi, in its discussion of Prophecy verses Philosophy, notes that seekers of the spiritual realm have several advantages over those who pursue a solely rational exploration of the intellect. Prophets perceive truths in both the spiritual and physical world, while also addressing issues that are pertinent to the most refined of people as well as the most simple. The Philosophic approach, however, focuses primarily on subjects of a worldly nature, and appeals only to those of trained and refined intellect. In addition, its great emphasis on critical thinking can degenerate to a cynical rejection of everything, leading its adherents to dissatisfaction and despair.
That being said, let me suggest several "rational" indications for the existence of the soul.
For one, consider the difference between a live body and one immediately after death. Both have the same biology and biochemistry. In both scenarios, the bodies and body parts are exactly the same. Why in one case do the eyes see, the ears hear, the mouth breathes etc., and in the other, the functions of the exact same organs cease? Lest one answer that the biological process of life has been terminated - biologically speaking, there's absolutely no difference. Furthermore, science itself has no explanation for the mystery of life. Rather, what's missing is the spiritual charge which energizes and animates the system. In this way, it can be compared to an appliance before and after losing its current. In both scenarios the system is exactly the same — what's missing is the charge.
Another indication rises from real-life experience. We've all been in a situation where in a room full of people, activity and commotion, we have an inexplicable, instinctual and nearly subconscious urge to suddenly turn around, only to find someone we don't even know to be staring at us. Our preoccupation with our own thoughts as well as the noise in the room would preclude our seeing or hearing the person we're being drawn to. Nor do we generally have this sensation, look around, and see no one such that one might argue the phenomenon is random. Rather, just as physical stimuli such as sound, smell and sight are perceived and communicated to us through their relative receptors, so too the intellectual and spiritual stimulus of one's focused attention is perceived and transmitted through the soul and mind to others.
Similar phenomena which operate on a more long-distance, far-reaching scale include those uncanny instances where we spontaneously think or dream of a person we haven't seen or thought of for a long time, or something that's going to happen, and then the person suddenly calls or appears in our lives, or the event actually happens. These and other such premonitions which are experienced universally also indicate that we all have some non-physical component which is aware of, in tune with, perceives and communicates to us information, "activity" and energy in the non-physical realm. In Judaism, this component is referred to as the soul. And by the way, the indisputably universal phenomenon of deja-vu is also understood by Judaism to be the soul's "recalling" to us events, situations, places and people from a previous reincarnation where the current re-interaction gives us the opportunity to repair and correct past misdeeds and shortcomings in order benefit the soul on its journey.
There are many more such indications, but I'll conclude with a recent personal experience. A saintly rabbi that I was once very close to, Rabbi Yosef Tzeinvert of blessed memory, passed away. On Shabbat around 10a.m. in the middle of concentrated prayer, I had a sudden thought of him which was clouded with a feeling of darkness and void. I was afraid to dwell on that feeling, and certainly didn't express it to anyone. Immediately after Shabbat ended I heard the terrible news that he had in fact passed away shortly before 10. At the funeral procession I shared this with another great rabbi, the Admor of Nadvorna-Bania, Rabbi Moshe Meir Lifer shlita, who told me that mystical sources in Judaism teach that when very holy and righteous people pass, they "inform" those who were close to them. A few days later, while visiting the mourners, I told them of my experience and what the Admor had said. Their immediate reaction was, "You have no idea how many people have told us this!" And right after that, a person who was sitting just next to me, whom I did not know, exclaimed, "I also came here to share the same experience!" and then he told his story too…