When he transformed into man-spider in the fox tv show, i think he actually did shoot webs from his arms. Don't recall if there was a more human version of him that did shoot webs as a super power over technology.
Like the story said, the movies didn't want to get into the development of them, so took the simpler explanation of "he's part spider, duh!" I did however, like the scene where the camera zoomed in on the barbs on his palms that let him grasp buildings, etc.
But I believe there have been versions of the character that had organic webbing, but it's been a while for me so I don't know what the latest ideas are.