The Spider-God's origins in the old world is unknown but it apparantly was content to reside in the Warp doing who knows what and being worshiped by nobody in particular.
Meanwhile on the mortal world, greenskins migrated throughout the world and mostly settled in the Badlands AKA Warhammer Eurasia (totally not a political statement, 'swear on our zog). A particular bunch of Goblins entered the woodlands of the Drakwald, essentially the Black Forest of Warhammer situated in future Empire territory. These Goblins found the forest to be mostly inhabited by giant Spiders of every size, from 'microscopic mite' to fucking 'four-story castle'.
The two initially were at constant war, with the spiders unable to defeat the Goblins, since greenskins release spores upon their death, leading to massive population growth where they are defeated. (although they weren't really trying, just eating what came to them). The Goblins, despite these ludicrous numbers, couldn't defeat the Spiders because they were massive apex predators and the Goblins were idiot spore-ape gits.
As time went on natural selection took place, and the dumber Goblins gave way to those clever enough to steal the young from their enemies to raise as mounts, giving way to the Forest Goblin Spider Riders as well as those smart enough to figure out which small spiders to eat, and which ones to avoid. The Goblins used sacrifices of Beastmen warbands captured by the Spider Riders as well as sacrifices both of their kin and with their own ritual bloodletting to appease the larger, more undefeatable spiders. Mutations began to take place among the Forest Goblins, resulting in extra arms, extra eyes, natural venom, and other spider-like traits appearing rarely among their population.
Eventually the Shamans of the Forest Goblins found a way to enter trances similar to those used to channel Gork and Mork (or was it Mork and Gork?) using poisonous spiders, either by eating them or filling their mouth with spiders, w hich proceeded to bite and pump their tongues full of venom. Either way, the minds of the Goblins were drug through the Warp and before the Spider-god itself. The Spider-god taught them dances which would prove kinship to the spiders of the world.
Some Shamans attempted to use these dances to communicate with the gigantic Arachnarok Spiders, the largest spiders in the world which previously had been able to wipe out life entirely in sections of the forest with their giant circular webs silently being woven around entire swathes of forest. These early attempts were met with predictable results: Goblin Shamans being wrapped up and liquified for a protein snack.
But the Shamans continued communing with the Spider-god, eventually becoming the #1 deity of the Forest Goblins with Gork and Mork/Mork and Gork as subordinate gods to its greatness. Effigies of spiders created with skulls adorned the Forest Goblin settlements, which were increasingly relocated to nearby Arachnarok lairs. Eventually the Arachnaroks began accepting the sacrifices of the Forest Goblins, and with the dances of the Shamans being echoed by the tribes they eventually began to see the Forest Goblins as their own young (note that since we're talking about spiders, that doesn't mean they won't eat any who get too close to the fangs or ensnared on a web), allowing them to crawl all over their bodies and build wooden structures held together with the Arachnarok's own web on their backs.
Since the Forest Goblins weren't important enough to even merit so much of a mention in End Times the Spider-God's participation was unknown.
Age Of Sigmar
The Spider-God survived into Age of Sigmar where it was confirmed to not only actually exist but also exist alongside a pantheon of other greenskin gods that are believed to be aspects of Gork and Mork/Mork and Gork/Gorkamorka. The Spiderfangs believe the Spider-God was a spider that bit the foot of Gorkamorka as he was rampaging through the Mortal Realms and became infused with the god's power, becoming an aspect of the god that the Spiderfang Grots worship. Though no one is entirely sure if it's a separate entity or not. Regardless it has been described as having a feral intelligence and its power can be felt in the places the Spiderfangs conquer and has been known to manifest its presence through the spiders that the Spiderfang Grots control (of which the sizes of the spiders can vary from a couple of inches to the size of dogs and bigger) and the Arachnaroks are believed to be its earthly avatars (somewhat like Spider-God Greater Daemons). The Spider-God also now has a following among assassins and other seedy outcasts of society dubbed the “Scuttling Queen”.
It is currently unknown if this Spider-God is even technically the same entity that existed in the World-That-Was.