I can go either or, it depends on the style of medium they have. Like in comic books mechanical fits but for the films organic is better. Personally I like mechanical web shooters better. Both can make sense if done right, USM introduced webbing in a great way and you can easily explain organic webs as a result of the mutation. Remember his body changed on a fundamental level, its not too hard to grasp that his arms developed muscles and other biological material that can mix different chemicals found naturally in the body to produce web fluids.
I read in an early issue Stan Lee just wrote a line that said "Peter Parker is the worlds formost expert on spiders" now remember Stan has no big grasp on science and usually just made up origins because they sounded cool. I also recall hearing that Stan originally wanted them to be organic but the company had issues with a teenage boy shooting goo out of his body. So there was never any real explanation for them to begin with so there's no real reason how he made them.
Now I like both version but what I hate is when they change it. Pick one way and stick with it, conviently changing his powers like they did in 616 is just absolutely retarded. I keep hoping for Peter to wake up and the past few years was just a dream