First of all, Spider-man doesn't have organic webs when the comics started. He got organic webs later in the comic series.
In the "Disassembled" storyline Parker undergoes a transformation that
results in the ability to produce organic web fluid from his wrists,
and is able to fire his webbing in much the same manner as his
artificial web-shooters. According to the new 2007 Spider-Man
handbook, Parker has grown spinnerets in his forearms that terminate
in small pores at the junction of his wrists. By pressing down with
his middle fingers to his palm, he causes the pores to open and the
spinnerets to eject the organic fluid with a force equal to or greater
than that of his web-shooters.
Click here for more details.
In Sam Raimi's trilogy of Spider-man, he did not follow the real Spider-man comics' story and skipped the Artificial Web-shooters.
By the way Sam Raimi's script is inspired by James Cameron scriptment, which took the idea of organic web-shooters for
Stevens's Failed Script. From 1985 there have been many scripts
written for the Spider-man . But James Cameron's Script got the most
attention and became the base of the 2002 film.[source]
But the 2012 The Amazing Spider-Man follows a similar path of the comics and they chose the artificial web-shooter for their movie.
Borrowed from my own answer from M&TV