The only good explanation for the webshooters is one that combines "organic" and "mechanical" into a coherent justification. A few people have already posted this and they are right!
To be clear – Stan Lee's mechanical shooters, and Sam Raimi's organic shooters are both ultimately for the convenience of economical storytelling. Neither actually "makes more sense" than the other. Arguing one over the other is extremely limited – it's fine to prefer one or the other, but if you genuinely believe one makes sense and the other doesn't, you are only kidding yourself. OP's question is a good one. When you think of a common household spider, you think of a few things – webs, bites, and eight legs. Peter Parker's powers relate to none of those things. Asking "why?" to that is less silly than not questioning it.
Everyone keeps making the same joke is distressing. "Because if it were organic it'd come from his butt." Well, not necessarily. It'd come from his torso. Maybe it milks out of his nipples, Greg. But again, he isn't Man-Spider... he's Spider-Man – and so, just as he doesn't have eight limbs or eyes, web fluid wouldn't necessarily need to come out of his butt just because that's how real spiders do it.
The best version, I think, comes from Brian Michael Bendis' Ultimate Spider-Man volume 1. In that version, Peter derives the formula from his father's work ,which he's able to finally crack in the rush of euphoria of his spider-mutation. It makes sense that his "spider-sense" would point him in a direction of creating webs –– just like a spider does not need to be taught how to weave web patterns, Peter learns this intuitively. It's both a mechanical invention, and a result of his organic powers.
It gets better once we learn that (in the Ultimate version) Dr. Parker's formula wasn't simply for a strong adhesive– it was part of the self-healing Venom suit that he was creating with Eddie Brock's dad.
So, in a sense, in this version the web fluid is organic (like a less advanced version of the Venom suit), and it is mechanical (the fluid and shooter are both simple tech that could conceivably be invented by a clever kid), and it could have only been discovered by Peter Parker (because it's his interpretation of his dad's formula while he's high on his new spider powers).
(supremely stan lee voice) HEY KIDS! THANKS FOR READING THIS MARVELOUS MESSAGE BOARD MISSIVE! HERE'S SOME FASCINATING FUN FACTS FOR YA!
The original Ditko/Lee comics called the web fluid "liquid cement" initially. If you think about it like that, it's easier to imagine the web fluid as something that isn't extraordinary in and of itself -- what's extraordinary is the creative way in which it's used as webbing.
One of the first things Peter tries to do was to capitalize on the idea and sell it. He's turned down because the webs disappear. (Which, incidentally, makes as little sense as the rest of it. Nobody else could see the potential here, or want to work on it to improve it? If you brought the web fluid on Shark Tank, you'd be a millionaire overnight).
Also, I hope everyone from this February 2018 thread has already seen and loved Into The Spider-Verse because searching for conversations about that is what led me here and to bumping this old bad boy.
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