As with other aspects of the game, as the GM you have a lot of leeway here. Just because the Adventure Path/Campaign/Monster Manual says that these creatures are spiders doesn't mean that they have to be. After all, all a monster is is "A block of stats with fluff" (Thanks @Grey Sage, I particularly liked that quote)
You can easily reskin spiders with creepy, net-wielding goblins (or gremlins, or kobolds, or lizardfolk, or whatever other race you feel fits best in your campaign)!
If you want them to maintain their low intelligence, they could be a tribe that's been so secluded from the rest of the world that they've mentally devolved. Their poison is now filth coating their weapons, and the web attacks are nets that they throw around PCs, or foul the ground with.
You can keep all of the stats of the spiders, and just re-skin them so you never have to say the word "spider". Instead of spitting webs they throw nets, instead of biting they swipe with their filthy weapons, and instead of climbing with 8 legs, they use nimble fingers to scale walls.
If your player really only has issues with spiders, and not other insects, you could also pick any other Giant insect you like to represent the spiders. Giant Beetles could shoot out a sticky liquid that traps your players while still having natural venom, and you could say the same for any number of large creepy-crawlies.
An exceptionally good alternative might be to make use of the sticky-footed lizards that are most often seen ridden by Drow. Smaller/younger versions (since the Drow mounts are Large) might be able to spit sticky saliva, or throw accumulated sticky from their feet, and lizard mouths are notoriously cess-pits of bacteria. The most important part to this process is just picking something that still fits in the campaign setting. Justifying your choice isn't all that hard, and the players are often lacking the background information necessary to really call out any 'mistakes' you might make.
As mentioned in several comments, you may want to consider re-creating the statblocks completely, so you don't accidentally read "spider" instead of "goblin" or "web" instead of "net" etc... The last thing you want is to reveal the charade, because at that point the whole gig is up and you'll have to do another, potentially more comprehensive, set of changes.