Very possible. Mental capacity and adrenaline can easily show how one can hold as much and so forth. So muscular structure and weight density can be something that genetically makes sense. Especially a radioactive spider. Only one is wall crawler. Can't really explain that. You have to consider the serious mutation if he had organic sacs. Despite his other abilities, his body should die from that sort of damage. That's just me over thinking it
Also I like the man-made as it shows how Peter is: a scientific scholar. Also plenty of his greatest defeats and victories have sometimes the factor of webbing. It makes him seem human, where Raimi's seems like a overnight wonder.
Yeah but I personally find it hugely implausible that a high-schooler could come up with this technology and somehow no one else has. That's at least as much of a ''wtf'' as having organic web shooters.
And considering then that at least one of his other abilities is completely inexplicable I think organic web-shooters have to get a pass.
I did like that the technological ones of the ASM films brought about scenarios where he had to repair them or find another way to tackle a situation but, as Difabio said, it didn't make the films any better. Also the Raimi films had that thing where his ability to use the shooters was influenced by his psychology so that worked in a similar way to malfunctioning or broken shooters.