I've always preferred web shooters over organic webs. The thought of webbing being inside a person's body is rather...disturbing.
In the movie, you saw a spider strand come out of Peter's upper spine. They could have made the web-wrists so that it has a tube connecting to his upper spine. Spiders keep all of their web fluid in their spine, so, not only would it have been "scientifically accurate", it would have made fans of the original comics happy. Everyone wins.