In the comics, way before he could naturally shoot webs; in the movies, he always uses them, or never uses them - it depends what movie your watching.
In the original Amazing Spider-Man comics, web shooters were a device Peter Parker created, in order to compliment his new-found abilities; his ability to shoot webs are almost always a result of this device.
The Sam Raimi trilogy, the three starring Tobey Maguire, was based on a script that was initially written by James Cameron. James Cameron had the brilliant idea that "organic web shooters" made more sense, and thus depicted them as such. This would later be adapted into the comic books, where in Peter Parker goes through more transformation, and grows organic web shooters; however, in the movie storyline, Spider-Man is given organic web shooters from the start. Regardless, in almost all depictions, the web-shooters are actual gadgets.
The gadget web-shooters make a return in 2012's The Amazing Spider-Man. Once again, this is a different universe to the Tobey Maguire Spider-Man and Homecoming. In Homecoming, this device is created by Peter and improved by Tony Stark1. Again, this is a completely differant universe - the "Marvel Cinematic Universe". It has no canonical connection to The Amazing Spider-Man movies, or the Spider-Man movies created by Sam Raimi.
1 Thanks to cde for clearing that up.