Do Vulcans have any beliefs that could be considered spiritual, beyond their obvious adherence to logic and empiricism?
They don't just believe... they have proof
In ST II, we see Spock place his Katra into McCoy.
In ST III, we see Spock's Katra placed into his regenerated body.
In Enterprise, Archer, T'Pau, and T'Les are all aware of the Katra of Surak by firsthand knowledge. Mention is made of polycrystaline vessels which could hold a katra. Further, there are priests with direct knowledge and possession of Katras besides their own.
In TAS: Yesteryear, Spock, under the false identity of Selek...
SAREK: My apologies, visitor. I regret you were witness to that unfortunate display of emotion on the part of my son.
SPOCK: In the family, all is silence. No more will be said of it. Live long and prosper, Sarek of Vulcan.
SAREK: Peace and long life. You are of my family?
SPOCK: My name is Selek, an humble cousin descended of T'Pel and Sasak. I am journeying to the family shrine to honour our gods.
SAREK: You have a long way to go. Will you break your journey with us for a while, Cousin?
In a couple of the novels, mention is made that certain groups (presumably Syranites, but not so called then) place Katra into stones, and are able to commune with them. This is apparently referenced into canon in Enterprise (Eps 83-85).
Similar mention is made in LUG Trek's book on the Vulcans.
The Archer incident (Ent Eps 83-85) establishes that, at least amongst a subset, belief in the soul is present in Vulcan culture. It also establishes that the Katra can appear to have a will of its own (as Surak refuses to enter T'Pau, staying in Archer.)
The resurrection of Spock is a strong proof as well.
Neither event, however, is public knowledge.
Sarek looking for Spock's Katra is indicative of an afterlife of some form - possibly proper entombment into a crystal, or being carried by a family priest.
And Enterprise establishes that there are Vulcan priests, since one appears in Kir'Shara.
http://www.chakoteya.net/Enterprise/83.htm Ent: The Forge
http://www.chakoteya.net/Enterprise/84.htm Ent: Awakening
http://www.chakoteya.net/Enterprise/85.htm Ent: Kir'Shara
The Vulcans held a number of spiritual beliefs, though little is known about the details. Their religious system was polytheistic. They also believed in the katra, the soul and consciousness of a person, which could be transferred psionically prior to death. (TAS: "Yesteryear"; Star Trek II: The Wrath of Khan; Star Trek III: The Search for Spock; TNG: "Gambit, Part I"; ENT: "The Forge", "Awakening")
While visiting Vulcan circa 8877 (on the Vulcan calendar, roughly equivalent to the Earth year 2237), Spock explained to his father that he was passing through ShiKahr during his journey to the family shrine, "to honor our Gods." (TAS: "Yesteryear")
This is if you consider this religion.
Reading the question, I remembered that Tuvok mentions Vulcan belief in an afterlife. Here is the relevant dialogue (VOY: "Innocence"):
TUVOK: Vulcans believe that a person's katra, what some might call a soul, continues to exist after the body dies.
ELANI: Do you believe that?
TUVOK: When I was younger, I accepted it without question. In recent years I have experienced doubts. I do believe there is more in each of us than science has yet explained.
This indicates a traditional belief that the katra separates from a deceased body to continue existing, similar to some human conceptions of the soul. Even Tuvok, an admitted skeptic of these beliefs, concedes the limitations of science in this matter. Therefore, most Vulcans are unlikely to have ruled out the supernatural as part of reality.
Taking this in conjunction with the events surrounding Surak's katra in Enterprise, I think it is fair to conclude that there is a spiritual side to Vulcan culture.
As the ST universe matured there were more and more examples of Vulcan meditation. That indicates to me their "religion" was more Eastern, i.e., Buddhist-like. Of course there is non-Eastern meditation, but, e.g., T'Pol seemed to follow classic Eastern meditation practice, not just contemplation or prayer. There was also plenty of indication that meditation was meant for seeking higher states, enlightenment. I would downplay the "gods," though. If we can accept the ST-Enterprise depiction of Vulcan-dom as the most mature and developed, I heard no overt talk of "gods," rather, a more Tibetan-style of Buddhism. Still, there seems plenty of the supernatural in the Vulcan spiritual world. But that supernatural (and the communion therewith) would seem to be attached to higher states of mind, again, an Eastern idea of spirituality.
Spock stated more than once he preferred the physical over the metaphysical, concrete facts over fairytales and he had a very low opinion of the human so called supreme being otherwise known as God.
Maybe Spock is an atheist but that doesn't mean all Vulcan are; I find it hard to picture T'Pay as a person with zero spiritual beliefs, considering who she is; I consider Kirk to be an agnostic; Dr. McCoy and Lt. Uhura were obviously true believers.
Especially Uhura in the closing scene of Bread and Circuses she reveals herself as a believer in Jesus Christ. when she explained that the "sunworshippers' were really son of God worshippers, not the sun in the sky! Perhaps the only time in Star Trek history that Jesus Christ is mentioned at all... and it hasn't happened since. Think about it.