Based on shots in this film, he gets bit some kind of unusual spider. Maybe it's radio-active, but who knows. There is a shot where he pulls out a gland or web from his neck where he got bit. My idea is that he uses the serum from this wound to create the material that he uses to fill the web shooters that he develops himself. This seems to fit the shots in the trailer that shows him testing his shooters. my feeling is that the wound secrets the enzyme to create the web, but his webshooters can obnly hold so much and eventually he has to prepare more.