This question might verge more into anthropology than history, depending on exactly what you mean by a “soul” and how speculative an answer you are willing to accept. The reason is that there is evidence of such beliefs before the development of writing, before the historical period. But archaeological evidence by itself is open to a lot of interpretation, while actual recorded words are more definite.

A common meaning of “soul” is some immaterial part or essence of a human being which is immortal, or at least survives past his death. By that loose definition, any belief in an afterlife requires a soul – some aspect of the human that continues past his physical death and then lives on in another plane, or is reborn back on Earth. So evidence of belief in an afterlife could be taken as a proxy for belief in souls.

Grave goods might be indicative of a belief in some sort of afterlife. Note that I don’t say “ritual burials”, because there are many possible secular explanations for those (for example, showing affection or respect for the deceased, not wanting their remains to be torn apart by wild animals). There is (disputed) evidence of intentional burials possibly as old as 300,000 years (Atapuerca in Spain), before the dawn of our own species. There are Neanderthal burials too, which appear intentional – Shanidar in Iraq, Kebara in Israel, Krapina in Croatia, for example. Some show cut marks on the bones, indicating ritual excarnation. In our own species, the earliest definitive evidence of intentional burials is at least 100,000 years old at Skhul and Jebel Qafzeh in Israel (for example, the double burial of mother and child with ochre painted bones). And then from around 40,000 years on, there is profuse evidence of elaborate burials from all over the world.

Beyond intentional burials, the presence of grave goods is stronger evidence of religious or spiritual belief in the afterlife. Grave goods are things that are interred along with the corpse. They could simply be items that the deceased used during his life – ornaments, tools, weapons, clothes. They could be animals, intentionally buried along with the deceased. Or food and drink.

This gets into speculation about religion and spirituality, which can't be avoided because that's the nature of your question. You can offer secular reasons for grave goods, for example, it’s “fitting” that a warrior be buried with his favorite bow, it would be an insult to his memory otherwise. But equally you could argue that perhaps it indicates that the warrior may need that bow in the afterlife. Grave goods like food or drink are difficult to explain otherwise, because they are not personal items like a weapon or jewelry might be, they are generic items for a particular use. An obvious explanation for their presence is that they'd be used to provide sustenance to the deceased in his journey into the afterlife, wherever that may be spent.

These arguments are speculative because we’re guessing at human intentions from limited evidence of behavior. They’re not proof of anything, they’re just guesses based on what we imagine about early humans. There’s no end to the speculation, you could argue that perhaps grave goods aren’t intended for the deceased, perhaps they’re to appease the gods, but then again perhaps the gods are being appeased on behalf of the dead to grant him favors in the afterlife.

There’s plenty of art from the Aurignacian on (~40,000 years and later) including cave paintings, figurines, decorative art, which have been interpreted with religious or spiritual symbolism. But again, it’s not possible to be sure. Are hunting scenes meant to invoke the blessings of the gods and favor future hunts, or did people just enjoy painting like we do? But then why paint them deep inside caves in hard to reach places where nobody lived? Does that have spiritual significance? What about obviously symbolic motifs – animals with human heads, weird critters that are a mix of body parts from different species? The famous Lion Man is over 40,000 years old, there’s plenty of carvings from Gobekli Tepe of very strange animals. Are these spirits or gods of some animistic religion, or were people just having fun?

Pretty much with the beginning of writing we have evidence of religion, but that doesn’t tell us how much further it goes. It seems very likely that the idea of souls predates writing, but that also makes it impossible to pinpoint any dates, because beliefs don’t fossilize, they can only be transmitted if they’re explicitly written down. There are plenty of lines of evidence that our distant ancestors had spiritual beliefs which may have included the notion of an afterlife. Grave goods are a particularly significant, in my opinion. But it can’t be proven.